imar.roimar.ro/~dtimotin/idei/proc.azuga.pdf · azuga { roma^nia, september 28-29, 2013 ... ky fan,...

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Printed by: Transilvania University Press B-dul Iuliu Maniu 41A,Brașov-România Phone: 0268-476050 E-mail:[email protected]

Editor: Nicolae Tița Transilvania University of Brașov E-mail:[email protected] or [email protected] All rights reserved Editorial board: I Cuculescu,Bucharest Univ., D.Pascali,Courant Inst.of New York Univ., I.Valușescu,Math.Institute of Romanian Academy. ISSN: 2360-2589

3

Cuprins

The papers presented to conference

1. A. G. Aksoy, J. M. Almira, On approximation schemes and compactness

5

2. A. G. Aksoy, S. Jin, The apple doesn’t fall far from the (metric) tree:equivalence of definitions

25

3. Aurelian Crăciunescu, Proprietes de factorisation pour multi-contractions

37

4. Maria Talpău Dimitriu, About tensor product stability of some operator ideals

61

5. Adela Sasu, Some special classes of approximation ideals

65

6. Nicolae Tița, Remarks on some classes of linear operators

70

7. Ilie Valusescu, Some remarks on the infinite-variate prediction

74

8. Dan Pascali, Old and new in the monogenic quaternion theory

103

4

Abstracts of some papers presented to conference

1. Anca Armășelu, A special class of ϕ,ΦL operators

122

2. Nicolae Tița, On some multilinear and bounded operators

123

Proceedings of the First Conference “Classical and Functional Analysis”

Azuga – Romania, september 28-29, 2013

Editor: Nicolae Tita

On approximation schemes and compactness

Asuman G. Aksoy and Jose M. Almira

1. Motivation

One of the basic notions in functional analysis is compactness. Its utility has be-come of fundamental importance after the appearance of Arzela-Ascoli’s Theorem[13], [14] especially pointing its use for the proof of existence results when inves-tigating the solutions of differential equations. Indeed, a key step for the proof ofconvergence in many algorithms is precisely to show that a certain set is compact,and many theorems have been produced to characterize compactness of subsets ofthe numerous function spaces and operator spaces that appear in functional analy-sis. The compactness of operators was also a main ingredient for the study of thesolutions of integral equations, and was indeed introduced by Hilbert in his studiesof the equations of Mathematical Physics. In particular, Hilbert and his studentSchmidth proved a very nice decomposition formula for all self-adjoint compactoperator T : H → H, where H is any separable Hilbert space: the spectral decom-position theorem. This theory was soon investigated and amplified to a beautifulset of results which we call nowadays Riesz theory (or Riesz-Schauder Theory) andis devoted to the study of operators S : X → X (where X denotes any complexBanach space) that can be expressed as S = λIX − T with λ 6= 0 (an scalar) andT : X → X, a compact operator. In such study, the spectral properties of theoperator T are essential and, in connection with these properties, it was soon dis-covered that some entropy and approximation quantities were of great importance(see, e.g., [21] for a detailed study of this connection). Compactness has also been afundamental concept for the development of other parts of Mathematical Analysis,such as Fixed Point Theory or Approximation Theory. Concretely, Brouwer’s fixedpoint theorem [18] asserts that every compact convex set K in Rn is a fixed pointspace, that is, if f : K → K is continuous, then f(x) = x for some x ∈ K (see [38, p.25] for a nice easy demonstration). On the other hand, Schauder’s fixed point the-orem [51], which has numerous applications in Mathematical Analysis, asserts thatevery convex set in a normed linear space is a fixed point space for compact maps(see also [16]). Among the results equivalent to Brouwer’s fixed point theorem, thetheorem of Knaster, Kuratowski and Mazurkiewicz (in short, KKM) [36] occupies aspecial place. Ky Fan, using KKM maps, was able to prove a best approximation

6 Asuman G. Aksoy and Jose M. Almira

theorem [29]. Later on, this concept was generalized by Khamsi to metric spacesetting by demonstrating a result which can be seen as an extension of Brouwerand Schauder’s fixed point theorems (see [35]). Finally, just to include in this sec-tion some results related to Approximation Theory, we would like to stand up thatcompactness of natural embeddings Y → X is, in fact, the main reason because,in many classical contexts, we can prove that approximation errors (with respect toarbitrary approximation schemes) and Fourier coefficients of functions that belongto the space Y , decay to zero with a certain prescribed behavior. This was recentlyproved by Almira and Oikhberg [12] and by Almira [8].

In this paper, we survey some results about the characterization of compactness inwhich the concept of approximation scheme has had a role. Concretely, in Section2 we present several results that were proved by the second author, jointly withLuther, a decade ago, when these authors were working on a very general theoryof approximation spaces [9], [10] (see also [31]) and, in Section 3, we introduce andshow the basic properties of a new concept of compactness, which was studied bythe first author in the eighties [1], [2], [3], [6], by using a generalized concept ofapproximation scheme and its associated Kolmogorov numbers, which generalizesthe classical concept of compactness.

2. Approximation schemes, approximation spaces and compactness

2.1. Preliminaries.

Definition 2.1. Given (X, ‖ · ‖) a quasi-Banach space, and A0 ⊂ A1 ⊂ . . . ⊂ An ⊂. . . ⊂ X an infinite chain of subsets of X, where all inclusions are strict, we say that(X, An) is an approximation scheme (or that (An) is an approximation scheme inX) if:

(A1) there exists a map K : N → N such that K(n) ≥ n and An +An ⊆ AK(n) forall n ∈ N,

(A2) λAn ⊂ An for all n ∈ N and all scalars λ,(A3)

⋃n∈N An is a dense subset of X.

We say that the approximation scheme (X, An) is nontrivial if An = AnX ( An+1

for all n, and we say that it is linear if An is a vector subspace of X for all n.

Approximation schemes were introduced in Banach space theory by Butzer andScherer in 1968 [20] and, independently, by Y. Brudnyi and N. Kruglyak under thename of “approximation families” in 1978 [19]. They were popularized by Pietschin his 1981 seminal paper [41], in which he introduced the approximation spaces

Arp(X,An) = x ∈ X : ‖x‖Ar

p= ‖E(x,An)∞n=0‖ℓp,r < ∞,

where

ℓp,r = an ∈ ℓ∞ : ‖an‖p,r =[ ∞∑

n=1

nrp−1(a∗n)p

] 1p

< ∞

On approximation schemes and compactness 7

denotes the so-called Lorentz sequence space (in particular, a∗n is the non-increasingrearrangement of an), (X, ‖·‖X) is a quasi-Banach space, and E(x,An) = infa∈An ‖x−a‖X .There were two main motivations for Pietsch’s study of approximation spaces. On

the one hand, the spaces Arp(X,An) form a scale which allows a natural interpreta-

tion of the so called central theorems in approximation theory as the appropriatetool for the classification of functions and operators in terms of their smoothness(compactness, respectively) properties, which crystallize with the property of mem-bership to one of these spaces (see, for example, [7], [25], [26], [42]). On the otherhand, he also detected a very nice parallelism between the theories of approximationspaces and interpolation spaces. In particular, he proved embedding, reiteration andrepresentation results for his approximation spaces.Simultaneously and also independently, Tita [60] studied, from 1971 on, for the

case of approximation of linear operators by finite rank operators, a similar concept,based on the use of symmetric norming functions Φ and the sequence spaces definedby them, SΦ = an : ∃ limn→∞ Φ(a∗1, a

∗2, · · · , a∗n, 0, 0, · · · ) and, later on, Almira

and Luther [9], [10] developed a theory for generalized approximation spaces via theuse of general sequence spaces S (that they named “admissible sequence spaces”)and defined approximation spaces as

A(X,S, An) = x ∈ X : ‖x‖A(X,S) = ‖E(x,An)‖S < ∞.Furthermore, this theory, which also includes the reiteration and representationtheorems, was developed by the authors without using any result from interpolationtheory. Admissibility of the sequence space S is just a technical imposition thatallows to prove that ‖x‖A(X,S) = ‖E(x,An)‖S defines a quasi-norm. This propertyis automatically satisfied by the sequence spaces S which contain all finite nullsequences and satisfy that, if bn ∈ S and |an| ≤ |bn| for all n, then an ∈ Sand ‖an‖S ≤ ‖bn‖S; if K(n) = n (see [9, Definition 3.2]). Other papers with asimilar spirit of generality have been written by Aksoy [1], [2], [3], [6], Tita [56] andPustylnik [46], [47]. Finally, a few other important references for people interestedon approximation spaces and/or approximation schemes are [11], [12], [22], [23],[24], [30], [39], [40], [57], [59], [60] and [58]. It is important to remark that, due tothe centrality of the concept of approximation scheme in approximation theory, theidea of defining approximation spaces is a quite natural one. Unfortunately, this hashad the negative effect that many unrelated people has thought on the same thingsat different places and different times, and some papers on this subject partiallyoverlap.Along this paper we will assume that all spaces appearing are normed, although

many of the results presented here also hold true in the quasi-normed setting.

2.2. Characterization of compactness with boundedly compact approxi-mation schemes and the Arzela-Ascoli Theorem. A first characterization ofcompactness in complete metric spaces was given by Hausdorff, who proved that


M is relatively compact in the complete metric space (X, d) if and only if for everyε > 0 there exists a finite ε-net for M (i.e., a finite set of points xksk=1 ⊆ X such

that M ⊆ ⋃Nk=1 Bd(xk, ε), where Bd(x, t) = y ∈ X : d(x, y) ≤ t). This result can

be reformulated as a characterization of compactness with the aid of approximationschemes as follows.

Theorem 2.2. Assume that (X, An) is an approximation scheme with An bound-edly compact for all n ∈ N, and let M ⊆ X. Then the following are equivalentclaims:

(i) M is a relatively compact subset of X(ii) M is a bounded subset of X and limn→∞E(M,An) = 0.

Furthermore, the implication (i) ⇒ (ii) holds true for arbitrary approximationschemes An.Proof. (i) ⇒ (ii) Assume that M ⊆ X is relatively compact. Then M is bounded

in X since MX

is bounded (compactness implies boundedness). We must showthat limn→∞E(M,An) = 0. Take ε > 0 and let x1, · · · , xN ⊆ X be an ε-net for M . Then, given x ∈ M , E(x,An) ≤ E(x − xk, An) + E(xk, An) ≤ ε +maxk=1,··· ,N E(xk, An) ≤ 2ε for n ≥ N0(ε), since limn→∞ maxk=1,··· ,N E(xk, An) = 0.Note that we have used nothing about An but the fact that

⋃n∈NAn is a dense

subset of X.(ii) ⇒ (i) Let ε > 0 be an arbitrary positive constant. By hypothesis, there exists

N0 = N0(ε) > 0 such that E(M,AN0) < ε/4. In particular, every x ∈ M admits adecomposition x = a(x) + y(x) with a(x) ∈ AN0 and ‖y(x)‖ = ‖x − a(x)‖ ≤ ε/2.Now, boundedness of M implies that there exists a constant C > ε such thatM ⊆ CUX , so that ‖a(x)‖ ≤ ‖x‖+ ‖y(x)‖ ≤ C + ε/2 ≤ 2C.

Let b1, b2, · · · , bs be a ε/2-net in AN0 ∩ 2CUX , which is a compact set since AN0

is boundedly compact. Given x ∈ M there exists i ≤ s such that ‖a(x)− bi‖ ≤ ε/2,so that

‖x− bi‖ ≤ ‖x− a(x)‖+ ‖a(x)− bi‖ ≤ ε,

which proves that b1, · · · , bs is a finite ε-net for M . Hausdorff’s theorem guaran-tees that M is relatively compact in X. Corollary 2.3. Assume that (X, An) is an approximation scheme with An bound-edly compact for all n ∈ N. A set M ⊆ X satisfies limn→∞ E(M,An) = 0 if and onlyif there exists M ′, a relatively compact subset of X, and a natural number N ∈ Nsuch that M ⊆ AN +M ′ is satisfied.

Proof. Assume that M ⊆ AN+M ′ with M ′ relatively compact in X. Then Theorem2.2 implies that E(M ′, An) ց 0. Take x ∈ M and n ∈ N, n ≥ N . Then thereexists a ∈ AN , y ∈ M ′ such that x = a+ y and

E(x,AK(n)) = E(a+ y, AK(n))

≤ E(a,An) + E(y, An)

= E(y, An) ≤ E(M ′, An),


so that E(M,AK(n)) ≤ E(M ′, An) for all n ≥ N , and E(M,An) ց 0.Let us now assume that E(M,An) ց 0. If M is a bounded subset of X then

Theorem 2.2 implies that M is relatively compact, so that we can take M ′ = Mand N = 0. On the other hand, if M is unbounded, then we can take N ∈ Nsuch that E(M,AN) ≤ 1/2 and define M ′ = y ∈ UX : exists x ∈ M and a ∈AN such that y = x − a. M ′ is obviously bounded and, if y = x − a ∈ M ′ witha ∈ AN , x ∈ M , then, for each n ≥ N ,

E(y, AK(n)) = E(x− a,AK(n))

≤ E(a,An) + E(x,An)

= E(x,An) ≤ E(M,An),

which proves that E(M ′, AK(n)) ≤ E(M,An) for all n ≥ N . Thus E(M ′, An) ց 0and Theorem 2.2 implies that M ′ is a relatively compact subset of X. Corollary 2.4 (Arzela-Ascoli). A set M ⊆ C[a, b] is relatively compact in C[a, b] ifand only if it is uniformly bounded and equicontinuous.

Proof. Let us consider the approximation scheme (C[a, b], Πn), where Πn denotesthe space of (algebraic) polynomials of degree ≤ n and let us assume that M isrelatively compact in C[a, b]. Then, Theorem 2.5 implies that M is a boundedsubset of C[a, b] (i.e., M is uniformly bounded, so that there exists C > 0 suchthat ‖f‖C[a,b] ≤ C for all f ∈ M) and E(M,Πn) ց 0. Let us show that M isequicontinuous.Given ε > 0 (without loss of generality we assume ε < C), there exists N ∈ N

such that E(M,ΠN) < ε/8. Furthermore, for all t, s ∈ [a, b] and all f ∈ M we have

|f(t)− f(s)| ≤ |f(t)− p(t)|+ |p(t)− p(s)|+ |p(s)− f(s)| for all p ∈ ΠN .

Hence, if we take p = p∗ ∈ ΠN such that ‖f − p∗‖C[a,b] ≤ 2E(f,ΠN), then

|f(t)− f(s)| ≤ 4E(f,ΠN)|+ |p∗(t)− p∗(s)| ≤ ε/2 + w(p∗, |t− s|),where w(h, δ) = sup|t−s|≤δ |h(t)−h(s)| denotes the modulus of continuity of the func-

tion h. Now, if p(t) = a0 + a1t+ · · ·+ aN tN ∈ ΠN , then both ‖p‖0 = max0≤k≤N |ak|

and ‖p‖1 = ‖p‖C[a,b] define a norm over the finite dimensional space ΠN , so that theyare equivalent norms. On the other hand, M being bounded, the norm of p∗ must becontrolled by a constant K > 0 (since ‖p∗‖ ≤ ‖p∗−f‖+‖f‖ ≤ ε/4+C ≤ 2C = K).This implies that we can assume max0≤k≤N |ak| ≤ K∗ for a certain constant K∗ > 0and hence

w(p∗, |t− s|) ≤ K∗N∑

k=0

w(φk, |t− s|); where φk(x) = xk, k = 0, 1, · · · , N.

(since w(ah1 + bh2, δ) ≤ max|a|, |b|(w(h1, δ) + w(h2, δ)) for all scalars a, b and

functions h1, h2, and p∗ =∑N

k=0 αkφk). In particular, we can choose δ = δ(ε) > 0such that |t − s| ≤ δ implies max0≤k≤N w(φk, |t − s|) ≤ ε

2K∗(N+1). This shows that


w(f, δ) ≤ ε for all f ∈ M , which is what we wanted to prove. To prove the otherimplication we can use Theorem 2.5 with An = Πn and the well known Jackson’sinequality for algebraic approximation E(f,Πn) ≤ Cw(f, 1

n+1), n = 0, 1, · · · .

In this section of the paper, we will concentrate our attention most of the time onlinear approximation schemes defined over Banach spaces X, since they are enoughfor the applications we mention explicitly here. In such a case it is known that all

sequence spaces ℓq(β) = an ⊂ R : ‖an‖ℓq(β) = (∑∞

n=0 bn|an|q)1q < ∞ are ad-

missible, so that, when dealing with these spaces we do not worry about the weightsβ = bn ⊂ [0,∞). Of course, if the approximation scheme is nonlinear and thespace ℓq(β) is not admissible for this approximation scheme, we still can talk aboutthe set A(X, An, ℓq(β)) and we will say that M is bounded in A(X, An, ℓq(β))whenever supf∈M ‖E(f, An‖ℓq(β) < ∞.

Theorem 2.5. Assume that (X, An) is an approximation scheme with An bound-edly compact for all n ∈ N. If q ∈ [1,∞] and M ⊆ X, then the following areequivalent statements:

(i) M is a relatively compact subset of X.(ii) There exists β = bn∞n=0 a sequence of nonnegative real numbers such that

‖β‖ℓq = ∞ and M is a bounded subset of A(X, An, ℓq(β)).Proof. We first show (i) ⇒ (ii). If M is relatively compact, then Theorem 2.2 provesthat αn = E(M,An) satisfies αn ∈ c0 and E(x,An) ≤ αn for all x ∈ M and alln ∈ N. Thus, if q = ∞, then supx∈M ‖E(x,M)‖ℓ∞( 1

αn) ≤ 1 and M is a bounded

subset of A(X, An, ℓ∞( 1αn)).

Let us now assume that q < ∞. Take nk a sequence of natural numbers suchthat αnk

≤ 2−k, k = 1, 2, · · · and consider the sequence β = bn defined by bnk= 1,

k = 1, 2, · · · , and bn = 12nαn

for n ∈ N \ nk∞k=1. Then ‖β‖qℓq ≥∑∞

k=1 bqnk

= ∞ and,for each x ∈ M ,

‖x‖qA(X,An,ℓq(β)) =∞∑

k=1

E(x,Ank)q +

∑

n∈N\nk∞k=1

E(x,An)q(

1

2nαn

)q

≤∞∑

k=1

2−kq +∑

n∈N\nk∞k=1

1

2qn≤ 3,

so that M is a bounded subset of A(X, An, ℓq(β)).Let us prove (ii) ⇒ (i). Let β = bn be a sequence of nonnegative real num-

bers such that b0 > 0 and ‖β‖ℓq = ∞. Assume that M is a bounded subset ofA(X, An, ℓq(β)). Then, b0 > 0 implies that M is also bounded in X. Further-more, given x ∈ M , we have that

E(x,An)‖bknk=0‖ℓq ≤ ‖bkE(x,Ak)nk=0‖ℓq≤ ‖bkE(x,Ak)∞k=0‖ℓq = ‖x‖A(X,An,ℓq(β)) ≤ C


for a certain constant C and all n ∈ N. This shows that E(M,An) ց 0, since‖β‖ℓq = ∞ and the estimation above holds for all x ∈ M . Theorem 2.2 implies thatM is a compact subset of X. Corollary 2.6. Assume that ‖β‖ℓq = ∞, where β = bn is a sequence of nonneg-ative real numbers and b0 > 0. If (X, An) is a linear approximation scheme withdimAn < ∞ for all n, the embedding A(X, An, ℓq(β)) → X is compact.

Proof. The linearity of An guarantees that ℓq(β) is an admissible sequence space forall β, so that A(X, An, ℓq(β)) is a Banach space and A(X, An, ℓq(β)) → X is anembedding. Now the Corollary is just a restatement of the implication (ii) ⇒ (i) inTheorem 2.5. Corollary 2.7. Assume that (Y, An) is a linear approximation scheme with dimAn <∞ for all n ∈ N, X is a Banach space, q ∈ [1,∞] and T ∈ L(X, Y ). Then the fol-lowing are equivalent statements:

(i) T ∈ K(X, Y ) (i.e., T is a compact operator).(ii) There exists a sequence of non-negative real numbers β = bn∞n=0 such that

b0 > 0, ‖β‖ℓq = ∞, and T ∈ L(X,A(Y, An, ℓq(β))).Proof. (i) ⇒ (ii). By hypothesis, T (UX) is relatively compact in Y , so that thereexists β = bn is a sequence of nonnegative real numbers such that ‖β‖ℓq =∞, b0 > 0, and T (UX) is a bounded subset of A(Y, An, ℓq(β)). Hence T ∈L(X,A(Y, An, ℓq(β))).(ii) ⇒ (i). This implication follows directly from the compactness of the embed-

ding A(Y, An, ℓq(β)) → Y . Theorem 2.5, in conjunction with the reiteration property of approximation spaces,

was used by Almira and Luther to prove a compactness criterium for subsets of gen-eralized approximation spaces and, as a corollary, a characterization of convergencein these spaces.To state these results it is necessary to introduce a little bit more notation. Con-

cretely, given β = bn∞n=0 a sequence of positive real numbers, we define the se-quence spaces

ℓq0(β) =

ℓq(β), whenever q < ∞c0(β) = an : limn→∞ anbn = 0, if q = +∞ .

These spaces appear here because, to use the reiteration property with an approxi-mation space A(X, An, S), it is necessary that

⋃n An be dense in A(X, An, S)

and, if S = ℓq(β) with ‖β‖ℓq = +∞, then the closure of⋃

nAn in A(X, An, ℓq(β))is A(X, An, ℓq0(β)).Theorem 2.8. Assume that ‖β‖ℓq = ∞, where β = bn is a sequence of nonneg-ative real numbers and b0 > 0, and let (X, An) be a linear approximation schemewith dimAn < ∞ for all n. The following assertions are equivalent:

(i) M is a relatively compact subset of A(X, An, ℓq0(β)).


(ii) There exists a sequence of nonnegative real numbers γ = an such that a0 >0, limn→∞ an = ∞, and M is a bounded subset of A(X, An, ℓq0(anbn)).

Theorem 2.9. Let us assume the hypotheses of Theorem 2.8. The sequence fn ⊆A(X, An, ℓq0(β)) is convergent in the norm of A(X, An, ℓq0(β)) if and only ifit is convergent in the norm of X and it forms a relatively compact subset ofA(X, An, ℓq0(β)).

2.3. Characterization of compactness with arbitrary linear approximationschemes and some applications. So far, we have imposed over An being bound-edly compact or, even more, being a finite dimensional linear space. Obviously, theseimpositions were necessary for our proofs, but it is also true that they are strongassumptions. Is it possible, for example, to give some compactness criterium byusing linear approximation schemes (X, An) if we allow dimAn = ∞? Obviously,in those cases the characterization of compactness should be more complicated sincebeing bounded in A(X, An, ℓq0(β)) will not be a sufficient condition for a boundedsubset of X in order to be relatively compact. The reason is simple: the unit ballof An, which is not relatively compact since An is infinite dimensional, is boundedin A(X, An, ℓq0(β)) for all β. Now, Almira and Luther [10] proved that, if M is abounded subset of A(X, An, ℓq0(β)), then compactness of M as a subset of X willfollow from some extra assumptions.

Theorem 2.10. Let (X, Ak) be a linear approximation scheme and assume thatthere exist linear projections Pk : X → X with Pk(X) = Ak for all k ∈ N, andsupk∈N ‖Pk‖ = K < ∞. Given M ⊆ X and q ∈ [1,∞], the following are equivalentstatements:

(i) M is relatively compact in X.(ii) Pk(M) is relatively compact in X for k = 1, 2 · · · and there exists β = bk ⊆

[0,∞) such that ‖β‖ℓq = +∞, b0 > 0 and M is bounded in A(X, Ak, ℓq(β)).

Proof. The implication (i) ⇒ (ii) is trivial. Indeed, it follows from (i) ⇒ (ii) inTheorem 2.2 -which holds true for arbitrary approximation schemes Ak- that, ifM is relatively compact in X then E(M,Ak) ց 0 and this is precisely what weneed for the existence of the sequence β with the desired properties. Furthermore, ifPk(fs)∞s=0 is an infinite sequence in Pk(M) then fs is also an infinite sequence inM , so that it admits a convergent subsequence fsi∞i=0. Obviously, Pk(fsi) is alsoconvergent since ‖Pk(fsi)−Pk(fsj)‖ ≤ ‖Pk‖‖fsi −fsj‖, which implies that Pk(fsi)is a Cauchy sequence. This proves that Pk(M) is relatively compact in X for all k.

Let us prove (ii) ⇒ (i). Let fm∞m=0 ⊆ M be an infinite sequence. We mustshow that fm contains a convergent subsequence. Let us define, for each k ∈ N,fk,m = Pk(fm). For each xk ∈ Ak we have that

‖fm − Pk(fm)‖ = ‖fm − xk − Pk(fm − xk)‖ ≤ ‖I − Pk‖‖fm − xk‖≤ (1 + ‖Pk‖)‖fm − xk‖.


Thus, if we take the infimum between the elements xk ∈ Ak, we get

‖fm − Pk(fm)‖ ≤ (1 + ‖Pk‖)E(fm, Ak) for all m, k ∈ N.

If we set ak =1 + ‖Pk‖‖biki=0‖ℓq

and ck = ‖biki=0‖ℓq , this inequality implies that

(1) ‖fm − Pk(fm)‖ ≤ ak(E(fm, Ak)ck + 1) for all m, k ∈ N.

Now, fixed k ∈ N, every subsequence of Pk(fm)∞m=0 contains a convergent subse-quence, since Pk(M) is relatively compact in X, by hypothesis. In particular, thesubsequence can be assumed to be of the form Pk(fm)m∈M0 (M0 an infinite subsetof N) and to satisfy the inequality(2)‖Pk(fs)−Pk(ft)‖ ≤ ak[(E(fs, Ak)+E(ft, Ak))ck+1]+ε for all t, s ∈ M0\[0,m0(ε)),

where ε > 0 can be arbitrarily small and m0 = m0(ε) may depend on ε.Using jointly the inequalities (1) and (2), and the triangle inequality,

‖ft − fs‖ ≤ ‖ft − Pk(ft)‖+ ‖Pk(ft)− Pk(fs)‖+ ‖fs − Pk(fs)‖,we have that

(3) ‖ft − fs‖ ≤ ak[2(E(fs, Ak) +E(ft, Ak))ck + 3] + ε for all t, s ∈ M0 \ [0,m0(ε)).

Obviously, ‖b‖ℓq = ∞ and supk∈N ‖Pk‖ = K < ∞ imply that ak ∈ c0. Further-more, the boudedness of M in A(X, Ak, ℓq(β)) implies that

(E(fm, Ak)ck)q = E(fm, Ak)

q‖biki=0‖qℓq ≤ ‖fm‖qA(X,Ak,ℓq(β)) ≤ Cq

for all m, k and a certain constant C > 0. Let us take n1 < n2 < · · · < ni < · · · asequence of natural numbers such that ani

(4C + 3) ≤ 2−i for all i = 1, 2, · · · , and anested sequence of infinite sets Mi ⊆ N, Mi+1 ⊆ Mi for all i, such that(4)‖ft−fs‖ ≤ ani

[2(E(fs, Ani)+E(ft, Ani

))cni+3]+2−i ≤ 2−i+2−i = 2−i+1 for all t, s ∈ Mi.

Then, if we choose m0 < m1 < · · · natural numbers such that mi ∈ Mi for all i, thesequence fmi

satisfies ‖fmi− fmj

‖ ≤ 2−i+1 for all j ≥ i, so that it is a Cauchysequence. This proves that M is relatively compact.

Obviously, if dimAk < ∞ for all k and M is a bounded subset of X, then Pk(M)is relatively compact for all k. In this sense, Theorem 2.10 is clearly a generalizationof Theorem 2.2. On the other hand, if X = H is a Hilbert space and Ak is a closedsubspace of H for all k, then the orthogonal projections Pk : H → H (Pk(H) = Ak)satisfy ‖Pk‖ = 1 for all k, so that Theorem 2.10 can be useful in this context forarbitrary linear approximation schemes. In fact, in their paper [10, Theorem 7.2],the authors used this result to give a new proof of Tjuriemskih’s lethargy theorem[61], [62] (see also [9], [53]):


Theorem 2.11 (Tjuriemskih). Let (X, An) be a nontrivial linear approximationscheme. Let εn ց 0 be a non-increasing sequence of positive numbers convergingto zero, and let us assume that at least one of the following two conditions is fulfilled:

(a) dimAk < ∞ for all k ∈ N.(b) X is a Hilbert space.

Then there exists f ∈ X such that E(f, Ak) = εk for all k ∈ N.

Furthermore, in [10, Theorem 7.11], Theorem 2.10 above were also used to provea compactness criterium, which generalizes Kolmogorov’s characterization of com-pactness in Lp(Rd) [37] (see also [32, Theorem 5]) and Simon’s characterization ofcompactness for Lp((0, T ), X) [52, Theorem 3.1], for the spaces

Lp(Rd, X) = f : Rd → X : f is measurable and‖f‖Lp(Rd,X) =

(∫

Rd

‖f(x)‖pXdx)p

< ∞,

where X is (any) Banach space.

Theorem 2.12. A bounded set M ⊆ Lp(Rd, X) is relatively compact in Lp(Rd, X)if and only if the following three conditions are satisfied:

(i) limk→∞∫‖x‖≥k

‖f(x)‖pXdx = 0 uniformly in f ∈ M .

(ii) The set ∫[a,b]

f(x)dx : f ∈ M ⊆ X is relatively compact for all a, b ∈ Rd

with a < b. (This means that a = (a1, · · · , ad), b = (b1, · · · , bd) satisfy ai < bifor all i, and [a, b] = [a1, b1]× · · · × [ad, bd]).

(iii) lim‖h‖→0 ‖f(·+ h)− f(·)‖p = 0 uniformly in f ∈ M .

A careful inspection of the proof of Theorem 2.10 reveals that the important stepsfor its arguments are the inequalities (1) and (2). This directly leads to introducethe following technical concept, and the reformulation of the result below it.

Definition 2.13 ((β, q)-condition). Let q ∈ [1,∞] and assume that β = bk∞k=0

is a sequence of positive numbers such that b0 > 0 and ‖β‖ℓq = ∞. Let (X, An)be a linear approximation scheme and let us assume that M ⊂ X. We say that Msatisfies the (β, q)-condition with respect to (X, An), if for every sequence fm ⊆M there exist ak ⊆ [0,∞) and sequences fk,m∞m=1 ⊆ X, k ∈ N, such that

(i) an ∈ c0(ii) ‖fm − fk,m‖X ≤ ak[E(fm, Ak)‖biki=0‖ℓq + 1] for all m, k ∈ N.(iii) For all k ∈ N, every subsequence of fk,m∞m=0 contains a subsequence fk,mm∈M0

(M0 been an infinite subset of N) such that

‖fk,s−fk,t‖X ≤ ak[(E(fs, Ak)+E(ft, Ak))‖biki=0‖ℓq+1] for all t, s ∈ M0\[0,m0(ε)),

where ε0 > 0 is arbitrarily small and m0 = m0(ε) may depend on ε.

Theorem 2.14. Let (X, Ak) be a linear approximation scheme and let q ∈ [1,∞]be fixed. The following are equivalent statements:

(i) M is relatively compact in X.


(ii) There exists β = bk ⊆ [0,∞) such that ‖β‖ℓq = +∞, b0 > 0 and M is abounded subset of A(X, Ak, ℓq(β)) which satisfies the (β, q)-condition withrespect to (X, An).

3. Generalized approximation schemes and Q-compactness

3.1. Preliminaries. A few examples.

Definition 3.1 (Generalized Approximation Scheme). Let X be a Banach space.For each n ∈ N, let Qn = Qn(X) be a family of subsets of X satisfying the followingconditions:

(GA1) 0 = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn ⊂ . . . .(GA2) λQn ⊂ Qn for all n ∈ N and all scalars λ.(GA3) Qn +Qm ⊆ Qn+m for every n,m ∈ N .

Then Q(X) = (Qn(X))n∈N is called a generalized approximation scheme on X. Weshall simply use Qn to denote Qn(X) if the context is clear.

Obviously, there are several important differences between this concept and Defi-nition 2.1 and, in fact, no one of these concepts includes the other one. We use herethe term “generalized” because the elements of Qn may be subsets of X (and notjust elements of X, as it was the case in Definition 2.1).Let us now consider a few important examples of generalized approximation

schemes:

1) The classical approximation schemes introduced in Pietsch in his seminal paper[41].

2) Qn = the set of all at-most-n-dimensional subspaces of any given Banach spaceX.

3) Let E be a Banach space and X = L(E); let Qn = Nn(E), where Nn(E) = theset of all n-nuclear maps on E. [42]

4) Let ak = (an)1+ 1

k , where (an) is a nuclear exponent sequence. Then Qn onX = L(E) can be defined as the set of all Λ∞(ak)-nuclear maps on E.[27]

We are now able to introduce Q-compact sets and operators:

Definition 3.2 (Generalized Kolmogorov Number). Let UX be the closed unit ballof X, Q(X) = (Qn(X))n∈N be a generalized approximation scheme on X, and Dbe a bounded subset of X. Then the nth generalized Kolmogorov number δn(D;Q)of D with respect to UX is defined by

(5) δn(D;Q) = infr > 0 : D ⊂ rUX + A for some A ∈ Qn(X).Assume that Y is a Banach space and T ∈ L(Y,X). The nth Kolmogorov numberδn(T ;Q) of T is defined as δn(T (UY );Q).

It follows that δn(T ;Q) forms a non-increasing sequence on non-negative numbers:

(6) ‖T‖ = δ0(T ;Q) ≥ δ1(T ;Q) ≥ · · · ≥ δn(T ;Q) ≥ 0.


Definition 3.3 (Q-compact set). Let D be a bounded subset of X. We say that Dis Q-compact if lim

nδn(D;Q) = 0.

Definition 3.4 (Q-Compact Operator). We say that T ∈ L(Y,X) is a Q-compactoperator if lim

nδn(T ;Q) = 0, i.e., T (UY ) is a Q-compact set.

Remark 3.5. If Q = An∞n=0 is a classical approximation scheme, then A ∈ An

means that A = An, so that, for any set D ⊆ X, δn(D; An) = E(D,An), since

δn(D; An) = infr : D ⊆ rUX + An= infr : E(x,An) ≤ r for all x ∈ D= sup

x∈DE(x,An) = E(D,An).

Hence, in this case D ⊆ X is An-compact if and only if E(D,An) ց 0 andTheorem 2.2 states that, if An is boundedly compact for all n, then D ⊆ X is rel-atively compact in X if and only if it is bounded in X and An-compact. Indeed,all theorems in Section 2 of the paper are also results about Q-compact sets or oper-ators. For example, Corollary 2.3 characterizes An-compactness of subsets of Xwhenever An is a boundedly compact approximation scheme on X.

Proposition 3.6. Let Q = Qn(X) be a generalized approximation scheme on Xand assume that all elements A ∈ Qn are cones (i.e., λA ⊆ A for all scalar λ),for n = 1, 2, · · · . If X is separable and δn(x;Q) ց 0 for all x ∈ X, then allrelatively compact subsets of X are Q-compact sets.

Proof. Let xn∞n=0 be a countable dense subset of X. For each n,m ∈ N, we takeAn,m ∈ Qm and an,m ∈ An,m such that

‖xn − an,m‖ ≤ 2E(xn, An,m) ≤ 3δm(xn, Q).

Then an,mn,m∈N is dense in X since limm→∞ δm(xn, Q) → 0 for all n ∈ Nand xn is dense in X. It follows that

⋃∞N=0 BN is dense in X, where BN =

spanan,mNn,m=1 is a linear subspace of X for all N . This obviously implies that,taking B0 = 0, the family Bn∞n=0 is a linear approximation scheme of X. Onthe other hand, it follows from (GA3) that, for each N , there exists K(N) ≥ N

and AK(N) ∈ QK(N) such that A1,1 + A1,2 + · · ·+ AN,N ⊆ AK(N). Furthermore, this

implies that BN ⊆ AK(N) since the sets An,m are cones. Hence

(7) δK(N)(M ;Q) ≤ E(M, AK(N)) ≤ E(M,BN ), for all M ⊆ X and all N ∈ N.We claim that if M is relatively compact in X, then E(M,Bn) ց 0. To provethis result, let us assume that the contrary is true. Then there exist yn∞n=1 ⊂ Mand c > 0 such that E(yn, Bn) > c for all n. The relative compactness of M impliesthat there exists a subsequence ynk

∞k=1 and y ∈ X such that limk→∞ ‖ynk−y‖ = 0.

Hence

E(ynk, Bnk

) ≤ E(ynk−y,Bnk

)+E(y,Bnk) ≤ ‖ynk

−y‖+E(y,Bnk) → 0( for k → ∞),


which contradicts c < E(ynk, Bnk

), k = 1, 2, · · · . It follows that E(M,Bn) ց 0and the inequalities (7) imply that M is Q-compact.

3.2. Q-Compactness Does Not Imply Compactness. In this section we showthat in Lp[0, 1], 2 ≤ p ≤ ∞, with a suitably defined approximation scheme, we canfind a Q-compact map which is not compact.Let [rn] be the space spanned by the Rademacher functions. It can be seen from

the Khinchin Inequality that

(8) ℓ2 ≈ [rn] ⊂ Lp[0, 1] for all 1 ≤ p ≤ ∞.

We define an approximation scheme An on Lp[0, 1] as follows:

(9) An = f ∈ Lp[0, 1] : f ∈ Lp+ 1n or simply An = Lp+ 1

n.

Lp+ 1n⊂ Lp+ 1

n+1gives us An ⊂ An+1. for n = 1, 2, . . . , and it is easily seen that

An + Am ⊂ An+m for n,m = 1, 2, . . . , and that λAn ⊂ An. Thus An is anapproximation scheme in the sense of Pietsch.Next we observe the existence of a projection

(10) P : Lp[0, 1] → Rp for p ≥ 2,

where Rp denotes the closure of the span of rn(t) in Lp[0, 1]. We know that for p ≥2, Lp[0, 1] ⊂ L2[0, 1]. Now R2 is a closed subspace of L2[0, 1] and P 2 : L2[0, 1] → R2

is an orthogonal projection onto R2. Then P = jP 2i, where i, j are isomorphismsshown in the Figure

L2i

P 2

R2j

P

Rp

Lp

Proposition 3.7. For p ≥ 2 the projection P : Lp[0, 1] → Rp is Q-compact but notcompact.

Proof. Let URp , ULp denote the closed unit balls of Rp and Lp respectively. It is easilyseen that P (ULp) ⊂ ‖P‖URp . But URp ⊂ CUR

P+ 1n

where C is a constant follows

from the Khinchin inequality. Therefore, P (ULp) ⊂ Lp+ 1n, which gives δn(P,Q) → 0.

To see that P is not a compact operator, observe that dimRp = ∞ and I − P isprojection with kernel Rp, so I − P is not a Fredholm operator. Therefore P is nota Riesz operator, but every compact operator is a Riesz operator. So P cannot bea compact operator.


Remark 3.8. Another example which proves that Q-compactness does not implycompactness: Take X = H a Hilbert space, Y ⊂ H an infinite dimensional closedsubspace such that dimH/Y = ∞, D = UY and T = PY : H → H the orthogonalprojection of H onto Y . Take An any nontrivial linear approximation scheme onH such that A0 = 0 and A1 = Y . Then D = T (UX) = UY is not relatively compactin X (so that T is not a compact operator) and δn(T, An) = δn(D; An) =E(D,An) = 0 for all n ≥ 1, so that T and D are An-compact.

3.3. Properties of Q-Compact Maps. Let A be the ideal defined as

(11) A = T ∈ L(X) : δn(T ;Q) → 0 as n → ∞,and let As denote the surjective hull of A, which is defined by

(12) As = T ∈ L(X) : δn(TQE1 ;Q) → 0 as n → ∞.where QE1 is a surjection of ℓ1I with QE1(Uℓ1I

) = UX .

Proposition 3.9.

i) Q-compact maps have separable range;ii) the uniform limit of Q-compact maps is Q-compact;iii) an ideal of Q-compact maps is equal to its surjective hull, i.e. A = As.

Proof. i) Follows from the definition. For ii) we first observe that δ0(T ;Q ≤ ‖T‖.Now suppose (Tn) is a sequence of Q-compact maps, and let T = lim

nTn. Then

δn(T ;Q) = δn(T − Tn + Tn;Q) ≤ δ0(T − Tn;Q) + δn(Tn;Q)

≤ ‖T − Tn‖+ δn(Tn;Q)(13)

which gives that T is Q-compact too.For iii), A ⊂ Ac follows from the fact that

(14) δn(TQE1 ;Q) ≤ δn(T ;Q)‖QE1‖ = δn(T ;Q);

on the other hand

(15) δn(TQE1 ;Q) ≤ δn(TQE1(Uℓ1I);Q) = δn(T ;Q);

gives the equality readily.

Remark 3.10. Let T be a linear mapping from a Banach space X into a Banachspace Y . According a classical theorem of Schauder ([28], p.485) an operator T ∈L(X, Y ) is compact if and only if its adjoint T ∗ ∈ L(Y ∗, X∗) is compact. UsingSchauder theorem Terzioglu [54] gave a representation theorem for compact maps.He proves that T ∈ L(X, Y ) is compact if and only if there is a sequence (un) ofcontinuous linear functionals on X with lim

n||un|| = 0 such that the inequality

||Tx|| ≤ supn

| < un, x > |


holds for every x ∈ X. In general Schauder type of theorem need not be true forQ-compact maps. However a result analogous to Terzioglu’s can be proved for Q-compact maps if one assumes both T and T ∗ are Q-compact. For details see [1].

3.4. Q-Compact Sets. We assume each An ∈ Qn(n ∈ N) is separable. It isimmediate from the definitions that Q-compact sets are separable and Q-compactmaps have separable range.

Definition 3.11 (Order-c0-sequence). A double sequence xn,kn,k∈N ⊂ X is saidto be an order-c0-sequence if the following hold:

(1) for every n ∈ N there exists an An ∈ Qn such that xn,k∞k=0 ⊂ An;(2) ‖xn,k‖ → 0 as n → ∞ uniformly in k.

Theorem 3.12. Suppose (X,Qn) is a generalized approximation scheme with setsAn ∈ Qn assumed to be solid (i.e, tAn ⊂ An for all t ∈ [0, 1]). Then a bounded subsetD of X is Q-compact if and only if there exists an order-c0-sequence xn,k∞k=0 ⊂ Xsuch that

(16) D ⊂ ∞∑

n=1

λnxn,k(n) : k(n)∞n=0 ⊆ N and∞∑

n=1

|λn| ≤ 1

.

Proof. Let D be Q-compact. Then δn(2D,Q) → 0 and so there exists n1 such that

(17) 2D ⊂ 1

4UX + An1 .

Since An1 is separable let x1,k∞k=0 be a countable dense subset of An1 ; then it iseasy to see that B1 = (2D+ 1

2UX)∩ x1,k∞k=0 6= ∅ (and is an infinite countable set)

and 2D ⊂ B1 +12UX .

Let D1 = (2D − B1) ∩ 12UX , where 2D − B1 is the ordinary vector difference.

Then D1 is a bounded set (since it is a subset of 12U) and given ǫ > 0 we get, by

the Q-compactness of 2D, that 2D − B1 ⊂ ǫUX + Am + An1 ⊂ ˜Am+n1 + ǫUX for

suitable m and suitable An1 ∈ Qn1 ,˜Am+n1 ∈ Qm+n1 ; this is true because B1 ⊂ An1

and λAn1 ∈ Qn1 for each λ. This shows that D1 is Q-compact and, as before, thereexists An2 ∈ Qn2 such that 2D1 ⊂ 1

8UX + An2 . Let x2,k∞k=0 be a dense subset of

An2 . Then

B2 = (2D1 +1

4UX) ∩ x2,k∞k=0 is infinite countable;(18)

2D1 ⊂ B2 +1

4UX ;(19)

D2 = (2D1 − B2) ∩1

4UX is Q-compact.(20)

Continuing this process we define

(21) Bm =

(2Dm−1 +

1

2mUX

)∩ xm,k∞k=0, xm,k∞k=0 dense in Anm ;


then 2Dm−1 ⊂ Bm + 12m

UX and we define

(22) Dm = (2Dm−1 − Bm) ∩1

2mUX .

Our construction gives for each d ∈ D, successively chosen bi ∈ Bi, i = 1, 2, . . . , ksuch that

(23) d−(1

2b1 +

1

22b2 + · · ·+ 1

2kbk

)∈ 2−kDk,

and since Dk ⊂ 2−kUX , it follows that

(24) d =∞∑

n=1

1

2nbn.

Since each bn = xn,k(n) for a suitable k(n) and since

bn ∈ Bn ⊂ 2Dn−1 +1

2nUX ⊂ 2 · 1

2n−1UX +

1

2nUX ⊂ 3

2n−2UX ,

it follows that ‖bn‖ → 0.In the reverse direction, suppose we have that for each n an An ∈ Qn and

xn,k∞k=0 ⊂ An with ‖xn,k‖ → 0 as n → ∞ uniformly in k and

(25) D ⊂∑

n

λnxn,k(n) :∞∑

n=0

|λn| ≤ 1 and k(n)∞n=0 ⊆ N

:= C.

Since for each c ∈ C we can write

(26) c =m∑

n=1

λnxn,k(n) +∞∑

n=m+1

λnxn,k(n) = u+ v,

where u ∈ λ1A1 + · · ·+ λmAm, our assumption on Qn and solidness of the An’s givethat u ∈ Am2 . Furthermore, given ǫ > 0 we may choose m such that ‖xn,k‖ < ǫ for

each k > m. Thus C ⊂ ǫU + Am2 and so δn(C,Q) → 0 as n → ∞, and therefore,also δn(D,Q) → 0.

Remark 3.13. Theorem 3.12 can be considered as an analogue of the Dieudonne-Schwartz lemma on compact sets in terms of standard Kolmogorov diameter. If onechooses Qn to be the at-most-n-dimensional subspaces of X one can show that Q-compactness of a bounded subset D coincides with the usual definition of compactnessof D

Remark 3.14. The first author and M.Nakamura have proven a similar theoremfor p-normed spaces, 0 ≤ p ≤ 1.

Next we give a characterization of Q-compact subsets of X via Q-compact mapsinto X.


Theorem 3.15. Assume (X,Qn) is a generalized approximation scheme on theBanach space X with each An ∈ Qn being a vector subspace of X. Then, a boundedsubset D of X is Q-compact if and only if D ⊂ T (UE) for a suitable Banach spaceE and a Q-compact map T on E into X.

Proof. We need only prove the “only if” part. Let D be Q-compact and let C denotethe closed absolute convex hull of D. Then that C is Q-compact is easily seen as

follows: each c ∈ C is of the form c =m∑

i=1

λidi withm∑

i=1

|λi| ≤ 1 and di ∈ D for each

i; give ǫ > 0, there exists N such that for all n ≥ N, δn(D,Q) < ǫ and equivalentlyD ⊂ ǫUX + An and obviously then C ⊂ ǫUX + An.Let XC denote the linear subspace of X spanned by the elements of C endowed

with the norm given by the gauge ( = Minkowski functional) µ of C. Then (XC , µC)is a Banach space (see, e.g., [48], [49]). Let E = (XC , µC). If T is the canonicalinjection of XC into X, then T (UE) = C ⊃ D and T is Q-compact.

Remark 3.16. Using order c0-sequences and associated sets Sm = m∑

n=1

λnxn, k(n) :

m∑

n=1

|λn| ≤ 1,one can define the ball measure of non-Q-compactness γ(D) of a

bounded set D in a Banach space X as γ(D,Q) = infr > 0 : D ⊂ ⋃x∈Sn

B(x, r).It is shown in [1] that

γ(D,Q) = limn

δ(D,Q).

Furthermore, if we denote by Qc the ideal of Q-compact maps, then the ideal varia-tion γQc(D) = infr > 0 : ∃E and T ∈ Qc(E,X) such that D ⊂ T (UE) + rUX =γ(D).

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A. G. AksoyDepartment of Mathematics. Claremont McKenna College.Claremont, CA, 91711, USA.email: [email protected]

J. M. AlmiraDepartamento de Matematicas. Universidad de Jaen.E.P.S. Linares, C/Alfonso X el Sabio, 2823700 Linares (Jaen) Spainemail: [email protected]




THE APPLE DOESN’T FALL FAR FROM THE (METRIC) TREE:EQUIVALENCE OF DEFINITIONS

Asuman G. AKSOY and Sixian JIN

Abstract

The equivalence of definitions for metric trees and for δ- hyperbolic spaces hasalways been assumed in the literature. Here we give a proof that these definitions areequivalent, and identify some fundamental geometric and metric properties of thesespaces and their relation to CAT(κ) spaces.

2000 Mathematics Subject Classification:54E35, 54E45, 54E50, , 47H09.Key words: Metric tree, Hyperbolic spaces, CAT(κ) spaces, Hyperconvex spaces.

1 Introduction

A metric space is a metric tree if and only if it is 0-hyperbolic and geodesic. In otherwords, a geodesic metric space is said to be a metric tree (or an R-tree, or T-tree) if itis 0-hyperbolic in the sense of Gromov that all of its geodesic triangles are isometric totripods. It is well known that every 0-hyperbolic metric space embeds isometrically intoa metric tree (see [14],[20]) and construction of metric trees is related to the asymptoticgeometry of hyperbolic spaces (see [11], [17]). Metric trees are not only described bydifferent names but are also given by different definitions. In the following, we state twowidely used definitions of a metric tree:

Definition 1.1. An R-tree is a metric space M such that for every x and y in M there is aunique arc between x and y and this arc is isometric to an interval in R (i.e., is a geodesicsegment).

Recall that for x, y ∈ M a geodesic segment from x to y denoted by [x, y] is the imageof an isometric embedding α : [a, b] → M such that α(a) = x and α(b) = y. A geodesicmetric space is a metric space in which every pair of points is joined by a (not necessarilyunique) geodesics.

26 Asuman G. AKSOY and Sixian JIN

Definition 1.2. An R-tree is a metric space M such that

(i) there is a unique geodesic segment denoted by [x, y] joining each pair of points xand y in M;

(ii) if [y, x] ∩ [x, z] = x ⇒ [y, x] ∪ [x, z] = [y, z] .

Condition (ii) above simply states that if two segments intersect in a single point thentheir union is a segment too. Note that Rn with the Euclidean metric satisfies the firstcondition. It fails, however, to satisfy the second condition.

The study of metric trees is motivated by many subdisciplines of mathematics [18],[34], biology, medicine and computer science. The relationship between metric trees andbiology and medicine stems from the construction of phylogenetic trees [33]; and conceptsof “string matching” in computer science are closely related with the structure of metrictrees [6]. Unlike metric trees, in an ordinary tree all the edges are assumed to have thesame length and therefore the metric is not often stressed. However, a metric tree is ageneralization of an ordinary tree that allows for different edge lengths. For example, aconnected graph without loops is a metric tree. Metric trees also arise naturally in thestudy of group isometries of hyperbolic spaces. For metric properties of trees we refer to[13]. Lastly, [31] and [32] explore the topological characterization of metric trees. For anoverview of geometry, topology, and group theory applications of metric trees,consult [7].For a complete discussion of these spaces and their relation to CAT (κ) spaces, see the wellknown monograph by Btidson and Haefliger [11]. Recall that a complete geodesic metricspace is said to be a CAT (κ) space (or a Hadamard space) if it is geodesically connectedand if every geodesic triangle in X is at least as ”thin” as its comparison triange in, re-spectively, the classical spherical space S2κ of curvature κ if κ > 0, the Euclidean plane ifκ = 0, and the classical hyperbolic space of curvature κ if κ < 0.

Given a metric d, we say that a point z is between x and y if d(x, y) = d(x, z)+d(z, y).It is not difficult to prove that in any metric space, the elements of a metric segment fromx to y are necessarily between x and y, and in a metric tree, the elements between x andy are the elements in the unique metric segment from x to y. Hence, if M is a metric treeand x, y ∈ M , then

[x, y] = z ∈ M : d(x, y) = d(x, z) + d(z, y).

The following is an example of a metric tree. For more examples see [3].

Example 1.1. (The Radial Metric) Define d : R2 × R2 → R+ by:

d(x, y) =

‖x− y‖ if x = λ y for some λ ∈ R,‖x‖+ ‖y‖ otherwise.

We can observe that the d is in fact a metric and that (R2, d) is a metric tree.

The Apple Doesn’t Fall Far From the (Metric) Tree 27

It is well known that any complete, simply connected Riemannian manifold havingnon-positive curvature is a CAT (0)-space. Other examples include the complex Hilbertball with the hyperbolic metric (see [21]), Euclidean buildings (see [12]) and classicalhyperbolic spaces. If a space is CAT (κ) for some κ < 0 then it is automatically CAT (0)-space. In particular, metric trees and a sub-class of CAT (0)-spaces, and we note thefollowing:

Proposition 1.2. If a metric space is CAT (κ) space for all κ then it is a metric tree.

For the proof of the above proposition see p. 159 of [11]. Note that if a Banach spaceis a CAT (κ) space for some κ then it is necessarily a Hilbert space and CAT (0). Theproperty that distinguishes the metric trees from the CAT (0) spaces is the fact that metrictrees are hyperconvex metric spaces. Properties of hyperconvex spaces and their relationto metric trees can be found in [1], [5], [25] and [27]. We refer to [8] for the properties ofmetric segments and to [2] and [4] for the basic properties of complete metric trees. Inthe following we list some of the properties of metric trees which will be used in the proofof Theorem 2.1.

1. (Uniform Convexity [3]). A metric tree M is uniformly convex.

2. (Projections are nonexpansive [4]). Metric projections on closed convex subsets of ametric trees are nonexpansive.

Property 1 above generalizes the classical Banach space notion of uniform convexityby defining modulus of convexity for geodesic metric spaces. Let C be a closed convexsubset (by convex we mean for all x, y ∈ C, we have [x, y] ⊂ C) of a metric tree M . If forevery point x ∈ M there exists a nearest point in C to x, and if this point is unique, wedenote this point by PC(x), and call the mapping PC the metric projection from M intoC. In Hilbert space, the metric projections on closed convex subsets are nonexpansive.In uniformly convex spaces, the metric projections are uniformly Lipschitzian. In fact,they are nonexpansive if and only if the space is Hilbert. Property 2 is remarkable in thiscontext and this result is not known in hyperconvex spaces. However, the fact that thenearest point projection onto convex subsets of metric trees is nonexpsansive also followsfrom the fact that this is true in the more general setting of CAT (0) spaces (see p. 177 of[11]).

A metric space (X, d) is said to have four-point-property if for each x, y, z, p ∈ X,

d(x, y) + d(z, p) ≤ max d(x, z) + d(y, p), d(x, p) + d(y, z)

holds.Four-point property characterizes metric trees [1] thus, it is natural extension charac-

terizes δ-hyperbolic spaces as seen in Definition1.3 below.

In the following we give three widely used definitions of δ-hyperbolic spaces with pos-itive δ and references to how these definitions are utilized in order to describe geometricproperties.


Definition 1.3. A metric space (X, d) is called δ − hyperbolic if for each x, y, z, p ∈X, d (x, y) + d (z, p) ≤ max d (x, z) + d (y, p) , d (x, p) + d (y, z)+ 2δ.

Definition 1.4. A metric space (X, d) is δ-hyperbolic if for all p, x, y, z ∈ X,

(x, z)p ≥ min(x, y)p , (y, z)p

− δ (1.1)

where (x, z)p = 12 (d (x, p) + d (z, p)− d (x, z)) is the Gromov product.

Definition 1.5. A geodesic metric space (X, d) is δ − hyperbolic if every geodesic triangleis δ − thin, i.e., given a geodesic triangle xyz ⊂ X, ∀a ∈ [x, y] , ∃b ∈ [x, z] ∪ [z, y] suchthat d (a, b) ≤ δ. [x, y] is the geodesic segment joint x, y.

Definition 1.4 is the original definition for δ-hyperbolic spaces from Gromov in [22],which depends on the notion of Gromov product. The Gromov product measures thefailure of the triangle inequality to be an equality. This definition appears in almost ev-ery paper where δ-hyperbolic spaces are discussed. Although one can provide a long listfrom our references we refer to [35] ,[9], [23], [24], [20] and [11]. The Gromov productenables one to define “convergence ” at infinity and by this convergence the boundary ofX, ∂X, can be defined. The metric on ∂X is the so called “visual metric” (see [9] and[11]). The advantage of Definition1.4 is that it facilitates the relationship between mapsof δ-hyperbolic spaces and maps of their boundary [9], [26].

Definition 1.3 is a generalization of famous four-point property for which δ = 0. Four-point property plays an important role in metric trees, for example, in [1], it is shown thata metric space is a metric tree if and only if it is complete, connected and satisfies thefour-point property. However, it is also well known that a complete geodesic metric spaceX is a CAT(0) if and only if it satisfies four-point property (see [11]). Furthermore, in[20] Godard proves that for a given metric space M , each Lipschitz-free spaces F (M) isisometric to a subspace of L1. This is equivalent to M satisfying four-point property, andthe fact that M isomerically embedds into a metric tree. The advantage of Definition 1.3is that we can write out the inequality directly by distance of the metric space instead ofby the Gromov product. In some cases if we construct a metric with the distance functionhaving a particular form; it is easier to deal with distance inequality than Gromov productinequality. For example, in [23], [24] Ibragimov provides a method to construct a Gromovhyperbolic space by ”hyperbolic filling” under a proper compact ultrametric space andsuch ”filling” of a space contains points which are metric balls in original ultrametric

space and is equipped with a distance function h (A,B) = 2 logdiam (A ∪B)√

diam (A) diam (B).

Note that the Definition 1.5 of δ-hyperbolic spaces requires that the underlying spaceis geodesic since it depends on geodesic triangles. Yet in [9] , Bonk and Schramm showthat any δ-hyperbolic space can be isometrically embedded into a geodesic δ-hyperbolic


space. Thus one has the freedom of using Definition1.5.

Furthermore, recall that we call X hyperbolic if it is δ-hyperbolic for some δ ≥ 0.Sometimes δ is refereed as a hyperbolicity constant for X. Besides any tree being 0-hyperbolic, any space of finite diameter, δ, is δ-hyperbolic and the hyperbolic plane H2

is (1

2log 3)-hyperbolic. In fact any simply connected Riemanian manifold with curvature

bounded above by some negative constant −κ2 < 0 is (1

2κlog 3)-hyperbolic (see[11]).

2 Main Results

Theorem 2.1. Definition 1.1 and Definition 1.2 of metric trees are equivalent.

Proof. Suppose M is a R-tree in the sense of Definition 1.1, and let x, y ∈ M. Then byDefinition 1, there is a unique arc joining x and y which is isometric to an interval in R.Hence it is a geodesic (i.e., metric) segment. So we may denote it by [x, y] . Thus we havedefined a unique metric segment [x, y] (= [y, x]) for each x, y ∈ M, so (i) holds. To seethat (ii) holds, suppose [y, x] ∩ [x, z] = x . Then, [y, x] ∪ [x, z] is an arc joining y andz; and by Definition1.1 it must be isometric to a real line interval. Therefore it must beprecisely the unique metric segment [y, z].

Now suppose M is a R-tree in the sense of Definition 1.2, and let x, y ∈ M. Then [x, y]is an arc joining x and y, and it is isometric with a real line interval. We must show thatthis is the only arc joining x and y.

Suppose A is an arc joining x and y, with A 6= [x, y] . By passing to a subarc, ifnecessary, we may without loss of generality, assume A ∩ [x, y] = x, y . Let P be anonexpansive projection of M onto [x, y] . Since P is continuous with P (x) = x andP (y) = y, clearly there must exist z1, z2 ∈ A\ x, y such that P (z1) 6= P (z2) . LetA1 denote the subarc of A joining z1 and z2. Fix z ∈ A1. If u ∈ A1 satisfies d (u, z) <d (z, P (z)) , then it must be the case that P (u) = P (z) . Here we use the fact that[x, P (z)]∩[P (z) , z] = P (z) ; hence by (ii) [x, z] = [x, P (z)]∪[P (z) , z] . Therefore, thereis an open neighborhood Nz of z such that u ∈ Nz ∩ A1 ⇒ P (u) = P (z) . The familyNzz∈A covers A1, so, by compactness of A1 there exist z1, · · ·, zn in A1 such that

A1 ⊂n⋃

i=1

Nzi . However, this implies P (z1) = P (z2) which is a contradiction. Therefore,

A = [x, y] , and since [x, y] is isometric to an interval in R, the conditions of Definition 1.1are fulfilled.

Remark 2.2. In the above proof we use the fact that the closest point projection ontoa closed metrically convex subset is noexpansive. Definition 1.2 is used in fixed pointtheory, mainly to investigate and see whether much of the known results for nonexpan-sive mappings remain valid in complete CAT (0) spaces with asymptotic centre type ofarguments used to overcome the lack of weak topology. For example it is shown that if


C is a nonempty connected bounded open subset of a complete CAT (0) space (M,d) andT : C → M is nonexpansive, then either

1. T has a fixed point in C, or

2. 0 < infd(x, T (x)) : x ∈ ∂C.Application of these to metrized graphs has led to ”topological” proofs of graph theoreticresults; for example refinement of the fixed edge theorem (see [27],[16], [28]). Definition1.1used to construct T-theory and its relation to tight spans (see [15]), and best approximationin R-trees (see [30]).

The equivalence of Definition 1.4 and Definition 1.5 are given in [22] for geodesic metricspaces. However, in the following we add one more definition and present more details.Here by saying Definition 1.4 is equivalent to Definition 1.5, we mean if a geodesic metricspace is δ -hyperbolic under one definition, then it is 3δ-hyperbolic under the other.

Theorem 2.3. Definition 1.3, Definition 1.4 and Definition 1.5 of δ-hyperbolic spaces areequivalent.

Proof. We suppose X be a geodesic Gromov δ − hyperbolic space below. We first showDefinition1.4 implies Definition1.3. By Definition1.4, we have

d (x, p)+d (y, p)−d (x, y) ≥ min d (x, p) + d (z, p)− d (x, z) , d (y, p) + d (z, p)− d (y, z)−2δ.

Without loss of generality we can suppose

d (x, p) + d (z, p)− d (x, z) ≥ d (y, p) + d (z, p)− d (y, z)

i.e.d (x, p) + d (y, z) ≥ d (y, p) + d (x, z) .

So we have d (x, p) + d (y, p) − d (x, y) ≥ d (y, p) + d (z, p) − d (y, z) − 2δ or equivalentlyd (x, p) + d (y, z) + 2δ ≥ d (z, p) + d (x, y).The same conclusion follows if we take

d (x, p) + d (z, p)− d (x, z) ≤ d (y, p) + d (z, p)− d (y, z)

and we have d (z, p) + d (x, y) ≤ max d (x, y) + d(x, z), d(x, p) + d(y, z)+ 2δ.

To show Definition1.3 implies Definition 1.4,without loss of generality, we suppose

d (x, z) + d (y, p) ≤ d (x, p) + d (y, z) .

So, we haved (x, y) + d (z, p) ≤ d (x, p) + d (y, z) + 2δ.

Then

d (y, p) + d (z, p)− d (y, z) ≤ d (x, p) + d (z, p)− d (x, z)

d (y, p) + d (z, p)− d (y, z)− 2δ ≤ d (x, p) + d (y, p)− d (x, y)


and we get

d (x, p)+d (y, p)−d (x, y) ≥ min d (y, p) + d (z, p)− d (y, z) , d (x, p) + d (z, p)− d (x, z)−2δ.

To prove equivalence of Definition 1.4 and Definition 1.5, we need the following property:

For any geodesic triangle xyz in metric space (M,d) we can find three points on eachside denoted by ax on [y, z] , ay on [x, z] and az on [x, y] such that

d (ax, y) = d (az, y) = (x, z)yd (az, x) = d (ay, x) = (y, z)xd (ay, z) = d (ax, z) = (x, y)z .

To show Definition1.5 implies Definition1.4, for any x, y, p ∈ X, we will show that for anyz ∈ X following holds:

(x, y)p ≥ min(x, z)p , (z, y)p

− 3δ.

Figure 1

By the above stated property, in triangle xyp we choose three points ap, ax and ay in[x, y] , [p, y] and [p, x] as shown in Figure 2 such that

d (y, ap) = (p, x)y = d (y, ax)

d (x, ap) = (p, y)x = d (x, ay)

d (p, ax) = (x, y)p = d (p, ay)

Without loss of generality we assume t ∈ [p, y] such that d (ap, t) ≤ δ andd (t, y) > d (ax, y) , then in triangle ytap,

d (t, y) < d (t, ap) + d (ap, y) = δ + d (ax, y)

so d (t, ax) < δ and the same conclusion follows if we suppose d (t, y) < d (ax, y).

Then for apaxt,

d (ap, ax) < d (t, ax) + d (ap, t) < 2δ.

For any z ∈ X, consider xyz and choose t1 ∈ [y, z] and t2 ∈ [x, z] such that d (ap, t1) andd (ap, t2) are the shortest distances from a to [y, z] and [x, z], therefore

min d (a, t1) , d (a, t2) ≤ δ.


Then looking at triangles papt1 and papt2 we have

min d (p, t1) , d (p, t2) ≤ min d (ap, t1) , d (ap, t2)+ d (p, ap)

≤ δ + d (p, ap) ≤ δ + d (p, ax) + d (ap, ax) ≤ 3δ + (x, y)p .

Since

(y, z)p =1

2(d (y, p) + d (z, p)− d (y, z)) =

1

2(d (y, p)− d (y, t1) + d (z, p)− d (z, t1))

by triangle inequality we have (y, z)p ≤ d (p, t1) and similarly for (x, z)p ≤ d (p, t2).

Then min((y, z)p , (x, z)p

)≤ min d (p, t1) , d (p, t2) ≤ 3δ + (x, y)p .

Figure 2

To show Definition 1.4 implies Definition 1.5, let xyz be a geodesic triangle andw ∈ [x, y] , see Figure 2. Let d (x, [y, z]) denote the shortest distance from x to the side[yz] , without loss of generality we assume (x, z)w ≤ (y, z)w . Then by Definition 1.4

(x, y)w ≥ min (x, z)w , (y, z)w − δ = (x, z)w − δ

which implies δ ≥ (x, z)w.

Next we consider the triangle xzw and find three points ax, az and aw on each side withthe previous property. Then

d (w, ax) = (x, z)wd (w, aw) ≥ d (w, [x, z]) .

Similarly, in xawz one can find three points bz, bw and baw on [w, aw] , [aw, z] and [z, w] ,which satisfy the previous property and we have d (w, baw) = (z, aw)w .

We assume (z, aw)w < (x, aw)w then d (w, ax) < d (w, baw) .

So,

δ ≥ min (z, aw)w , (x, aw)w − (x, z)w = (z, aw)w − (x, z)w = d (baw , w)− d (ax, w)

=d (ax, baw) = (aw, z)w − (x, z)w=1

2 (d (aw, w) + d (x, z)− d (aw, z)− d (x,w))

=12 (d (aw, x) + d (w, bz) + d (aw, bz)− d (w, az)− d (aw, x))

=12 (d (ax, baw) + d (aw, bw))

which implies d (ax, baw) = d (aw, bw) .

Thus, d (w, aw) = d (w, bz) + d (bz, aw) = d (w, baw) + d (aw, bw) = d (w, ax) + 2d (aw, bw) ≤(x, z)w + 2δ.

Then d (w, [x, z]) ≤ d (w, aw) ≤ (x, z)w + 2δ ≤ 3δ.


Remark 2.4. In [10] Bonk and Foertsch use the inequality (1.1) repeatedly to define a newspace ACu (κ)-space by introducing the notion of upper curvature bounds for Gromovhyperbolic spaces. This space is equivalent to a δ-hyperbolic space and furthermore itestablishes a precise relationship between CAT(κ) spaces and δ-hyperbolic spaces. Itis well known that any CAT(κ) space with negative κ is a δ-hyperbolic space for someδ. In [10] it is shown that a CAT(κ) space with negative κ is just an ACu (κ)-space.Moreover, following the arguments in [10], Fournier, Ismail and Vigneron in [19] computean approximate value for δ .

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Asuman Guven AKSOYClaremont McKenna CollegeDepartment of MathematicsClaremont, CA 91711, USAE-mail: [email protected]

Sixian JINClaremont Graduate UniversityDepartment of MathematicsClaremont, CA, 91711, USAE-mail: [email protected]




Proprietes de factorisation pour multi-contractions

Aurelian Craciunescu

Abstract. On consider les systemes commutatifs de contractions qui admettentun calcul fonctionnel faible∗ continu sur H∞, et une dilatation isometrique. Dansl’hypothese de dominance du spectre de Harte on va obtenir (dans une condition su-plimentaire) la propriete (Aℵ0

) pour ces systemes. Dans l’hypothese de dominancedu spectre droit de Harte on obtient la propriete (A1,ℵ0

) pour l’ampliation d’ordredeux du systeme. Pour cela est introduit un nouvel ensemble d’approximation basesur l’ensemble utilise par J. Eschmeier en [9], mais dans lequel sont ”melanges” lesconditions d’annulation.

Keywords: sous-espace invariant, algebre duale, systeme de contractions, spectre deHarte, proprietes de factorisations.

MSC(2000): Primary 47A15; Secondary 47A13, 47L45

1 Introduction

La methode d’approximation successive (utilisee initialement pour montrer que tout operateursous-normal possede des sous-espace invariant non-trivial (voir [5])) a introduit le concept d’algebreduale. L’interet des techniques d’algebres duales depasse le cadre du probleme des sous-espacesinvariants parce que la methode, introduite par Scott Brown en 1978, pose encore deux problemesessentiels: le probleme de l’existence d’un “bon”calcul fonctionnel de type H∞ et “le probleme dela factorisation” dans le predual d’une algebre duale. Dans ce contexte sont definies les proprietesde factorisation (Ap,q) (p, q ∈ N ∪ ℵ0) pour une algebre duale, de maniere qu’on avait reussia caracteriser les proprietes (Aℵ0

), respectivement (A1,ℵ0), d’une algebre duale engendree par

une contraction, a l’aide du concept d’ensemble d’approximation et a l’aide d’une propriete dedominance du spectre. Dans cette direction, G. Robel montre ([13]) que toute contraction pourlaquelle le probleme de calcul fonctionnel a une reponse positive et qui a le spectre essentiel“suffisamment” riche (dominant, dans la terminologie usuelle) a la propriete X0,1. Mais, d’apresun resultat de Brown-Foias-Pearcy la propriete X0,1 est equivalente a la propriete (Aℵ0

) (voir [6]),donc toute contraction a spectre essentiel dominant a un ensemble riche de sous-espaces invariants.En 1988 B. Prunaru a montre que une contraction, absolument continue, a spectre dominant a lapropriete (A1) (voir [12]), resultat generalise, independamment, par H. Bercovici et B. Chevreauqui ont montre que, en fait, les classes A et A1 concident. Dans le meme anne, B. Chevreau,G. Exner, et C. Pearcy ont reussi a caracteriser la classe A1,ℵ0

en utilisant un nouvel ensemble

Proprietes de factorisation pour multi-contractions 38

d’approximation (note Erθ ) et une nouvelle methode technique permettant de combiner la methode

initiale avec “l’appareil” fonctionnel provenant de la partie unitaire de l’extension coisometriqueminimale de la contraction (voir [8]).

Nous avons considere dans ce travail le cas d’une algebre duale engendree par un systemecommutatif de contractions, qui admet une dilatation isometrique et pour lequel il existe un calculfonctionnel, d’algebres de Banach, sur H∞(Dm), failbe∗ continu prolongeant le calcul polynomial.En utilisant deux resultats obtenus par B. Chevreau (Th.2.7 et Th.2.8) qui generalisent, dans lecas des systemes de contractions, les resultats de G. Robel, respectivement Brown-Foias-Pearcy, onva montrer que si un tel systeme a le spectre droit de Harte dominant alors son ampliation d’ordredeux (ou meme le systeme s’il est de type K0,.) a la propriete (A1,ℵ0

) (Th.5.1). Si le spectrede Harte est dominant on va obtenir la propriete (A1) pour l’ampliation d’ordre huit (ou (Aℵ0

)pour le systeme s’il est de type K0,0) (Th.4.3). Pour cela est introduit l’ensemble d’approximationZθ (semblable avec l’ensemble Er

θ utilise en [9] mais dans lequel les conditions d’annulations sont”melangees”) qui permet l’amelioration d’une approximation donnee dans certaines conditions(Section 3).

2 Preliminaires

Soit H un espace de Hilbert complexe separable de dimension infinie, (dont le produit scalairesera note <,>), L(H) l’algebre des operateurs lineaires et bornes sur H et C1(H) l’espace desoperateurs a trace finie. Une algebre duale sur H est une sous-algebre de L(H) unitaire et fermeepar rapport a la topologie faible∗ (notee f∗) donnee par la dualite L(H) = (C1(H))

∗. On remarque

que, si A est une algebre duale sur H alors A = (QA)∗ou QA := C1(H)/⊥A (nomme le predual de

A), et ⊥A := K ∈ C1(H) / ∀A ∈ A ⇒< K,A >= 0 (l’application L(H) × C1(H) ∋ (T,K) 7→<T,K >:= tr(TK) ∈ C note la forme bilineaire associee a la trace (notee tr)). Si x, y ∈ H on note[x⊗ y] la classe (dans QA) determinee par l’operateur (x⊗ y)u :=< u, y > x, (u ∈ H).

Nous rappelons que, si 1 ≤ p, q ≤ ℵ0 sont deux cardinaux, on dit que l’algebre duale A a lapropriete (Ap,q) (ou (Ap) si p = q) si pour toute matrice ([Li,j ]) 0 ≤ i < p

0 ≤ j < q

a coefficients dans QA,

il existe des suites (xi)0≤i<p, (yj)0≤j<q dans H telles que [Li,j ] = [xi ⊗ yj ], 0 ≤ i < p, 0 ≤ j < q.Rappelons aussi qu’etant donne un entier n ≥ 1, on dit qu’une algebre duale A a la propriete

(A1/n) si pour tout element [L] de QA il existe des suites (xi)1≤i≤n et (yi)1≤i≤n dans H, tel que

[L] =n∑

i=1

[xi ⊗ yi].

Pour un entier m ≥ 1 on note H∞(Dm) l’algebre de Banach des fonctions analytiques borneessur Dm ⊂ Cm (D note le disque unite ouvert de C) et a valeurs complexes, munie de la norme dusupremum (||f ||∞ := sup

z∈Dm

|f(z)|).Un sous-ensemble σ de Cm est dominant dans Dm si ||f ||∞ = sup

z∈σ∩Dm

|f(z)| pour tous f ∈H∞(Dm).

L’espace H∞(Dm) est un sous-espace f∗−ferme de L∞(Dm) relativement a la dualite <L1(Dm), L∞(Dm) > (par rapport a la mesure Lebesgue 2n−dimensionnelle). Nous notons Q :=L1(Dm)/⊥H∞(Dm) le predual de H∞(Dm), ou ⊥H∞(Dm) est son preanihilateur (voir [7], [10]ou [15]).

Si T = (T1, . . . , Tm) ∈ (L(H))m

est un systeme commutatif de contractions sur H (nomme parla suite m−uplet commutatif de contractions ou simplement, multi-contraction), nous notons AT

l’algebre duale engendree par T . On dit que T a la propriete (Ap,q) (resp. (A1/n)) si l’algebre AT

a cette propriete. Le systeme T est dit de classe K0,. (resp. K.,0) si le produit T1T2 . . . Tm (resp.


T ∗1 T

∗2 . . . T ∗

m)est de classe C0,., c’est-a-dire, limn→∞

||Tn1 T

n2 . . . Tn

mx|| = 0, pour tous x ∈ H. La classe

K0,0 est la l’intersection K0,. ∩K.,0.Le systeme T est dit absolument continu s’il existe un calcul fonctionnel d’algebres de Banach

ΦT : H∞(Dm) → AT , h 7→ ΦT (h) =: h(T )

f∗−continu prolongeant le calcul fonctionnel polynomial. Nous allons dire que T est de classe As’il est absolument continu et ΦT est isometrique.

On note Lat(T ) le treillis des sous-espaces lineaires fermes Y de H tels que TjY ⊂ Y, j = 1,m(un tel sous-espace est dit invariant pour le systeme T ).

Si λ ∈ Dm on note Eλ la fonctionnelle d’evaluation en λ, c’est-a-dire la fonctionnelle (f∗−continue,donc identifiee a un element de Q) definie sur H∞(Dm) par < h,Eλ >:= h(λ).

Pour un systeme commutatif, absolument continu, T ∈ (L(H))m

et x, y ∈ H on note x ⋄T y lafonctionnelle f∗−continue (donc element de Q) definie par:

x ⋄T y : H∞(Dm) → C, < h, x ⋄T y >:=< h(T )x, y >, (h ∈ H∞(Dm))

S’il existe λ ∈ Dm et des vecteurs x, y dans H tel que:

(1) Eλ = x ⋄T y

alors avec Y :=∨h(T )x : h ∈ Ker Eλ nous avons < x, y >= 1, y ⊥ Y et donc Y ou Cx est

invariant non-trivial.De plus, s’il existe λ ∈ Dm et des suites (xn)n≥0, (yn)n≥0 dans H telles que:

(2) δk,jEλ = xk ⋄T yj , (k, j ∈ N)

(ou δk,j note le symbole de Kronecker), notant

Y0 :=∨

h(T )xn : h ∈ Ker Eλ, n ∈ N

Y1 :=∨

h(T )xn : h ∈ H∞(Dm), n ∈ N

nous aurons dimY1/Y0 = ∞ et l’ensemble Y ∈ Lat(T ) : Y0 ⊂ Y ⊂ Y1 est infini (voir [1]).La resolution des equations (1) et (2) se fait par la methode des approximations successives.

Le principe de cette methode est d’ameliorer une approximation donnee d’un element du predualQA (A etant une algebre duale) avec des elements de type [x ⊗ y]. Un role important dans cettemethode est joue par l’ensemble d’approximation defini, pour une algebre duale A, de la faconsuivante:

Definition 2.1. Soit θ ∈ [0, 1). L’ensemble Xθ(A) est constitue des elements [L] ∈ QA pourlesquels il existe des suites (xn)n≥0, (yn)n≥0 dans la boule unite fermee de H telles que:

limn→∞

||[L]− [xn ⊗ yn]|| ≤ θ, (2.1)

limn→∞

||[xn ⊗ w]|| = 0, (w ∈ H) (2.2)

limn→∞

||[w ⊗ yn]|| = 0, (w ∈ H) (2.3)


Les conditions (2.2) et (2.3) de la definition anterieure s’appellent conditions d’annulation (oude disparition), la demonstration de telles conditions etant l’objet de la premiere partie de cetravail.

L’ensemble Xθ(A) est un sous-ensemble absolument convexe et ferme de QA (voir Prop. 5.12de [7]), son role et son importance dans l’etude des proprietes (Ap,q) etant mis en evidence parle resultat suivant (voir [7], Th.5.13) (rappelons que pour un espace norme, X, et γ > 0 on note(X)γ la boule unite fermee centree en origine et de rayon γ):

Proposition 2.2. Soit A une algebre duale. S’il existe γ > θ ≥ 0 tel que (QA)γ ⊂ Xθ(A) alors Aa la propriete (Aℵ0

).

L’hypothese de la proposition anterieure definit la propriete Xθ,γ pour l’algebre A (ou pour lesysteme d’operateurs T dans le cas A = AT ). L’un des resultats principaux de ce travail dit quesi T est un systeme K0,0 a spectre de Harte dominant alors il a la propriete X0,1 (et donc Aℵ0

).Le spectre de Harte sera defini plus bas, mais d’abord nous allons faire un court commentaire surl’ensemble d’approximation dans le cas de l’algebre duale engendree par un systeme commutatif decontractions de classe A. Soit T un tel systeme et ϕT : QT → Q le preadjoint de ΦT (qui est uneisometrie aussi). Alors ϕT ([x ⊗ y]) = x ⋄T y, (x, y ∈ H) et ϕT (Xθ(AT )) coincide avec l’ensembledes elements [f ] de Q pour lesquels il existe des suites (xn)n≥0, (yn)n≥0 dans la boule unite fermede H telles que:

i) limn→∞

||[f ]− xn ⋄T yn|| ≤ θ, (2.1)′

ii) limn→∞

||xn ⋄T w|| = 0, (w ∈ H), (2.2)′

iii) limn→∞

||w ⋄T yn|| = 0, (w ∈ H). (2.3)′

On va noter Xθ(T ) ce dernier ensemble. Il est facile de voir que T a la propriete Xθ,γ si et seulementsi Xθ(T ) ⊃ (Q)γ .

Pour T = (T1, . . . , Tm) ∈ (L(H))m, (TiTj = TjTi) on considere les applications:

ΘT : H → Hm, x 7→ (T1x, . . . , Tmx)

respectivementΩT : Hm → H, (x1, . . . , xm) 7→ T1x1 + . . .+ Tmxm

On dit que λ = (λ1, . . . , λm) ∈ Cm apartient au spectre de Harte de T (note par σH(T )) siΘT−λ n’est pas borne inferieurement ou ΩT−λ n’est pas surjective. On dit que λ est dans le spectreessentiel gauche de Harte (note par σH

le (T )), si ΘT−λ a l’image non-fermee ou le noyau est dedimension infinie. Le spectre essentiel droit de Harte (σH

re(T )) est defini comme l’ensemble de tousλ ∈ Cm pour lesquels ΘT−λ a l’image de codimension infinie. Le spectre essentiel de Harte est lareunion σH

e (T ) = σHle (T ) ∪ σH

re(T ).Rappelons aussi que λ ∈ σH

le (T ) si et seulement s’il existe une suite orthonorme dansH, (xn)n≥0,

telles que limn→∞

||(Tj − λj)xn|| = 0, (j = 1,m), et λ ∈ σHre(T ) si et seulement si λ ∈ σH

le (T∗), ou

T ∗ := (T ∗1 , . . . , T

∗m) est le systeme adjoint. Si on note Ker (T ) :=

m⋂j=1

Ker (Tj), σp(T ) := λ ∈

Cm : Ker (T − λ) 6= (0) et Λ(T ) := λ ∈ σp(T ) : 1 ≤ dim (Ker (T − λ)) < ∞, alors:

σH(T ) = σHe (T ) ∪ Λ(T ) ∪ Λ(T ∗)

ou, pour un ensemble Λ ⊂ Cm, Λ note l’ensemble des conjugues des elements d’ensemble Λ (c’est-

a-dire, λ = (λ1, . . . , λm) ∈ Λ si et seulement si λ := (λ1, . . . , λm) ∈ Λ).


Si le systeme T est absolument continu et λ ∈ σp(T ) alors il est facile de voir que Eλ =u ⋄T u pour tout u ∈ Ker(T − λ). Une telle relation reste vraie aussi pour tous les combinaisons

absolument convexes d’elements de type Eλ avec λ ∈ σp(T ) (ou λ ∈ σp(T ∗)). Ce dernier resultatest une consequence du lemme suivant (voir [4], pg. 18 - 20), enonce ici dans le cas des espaces deHilbert:

Lemme 2.3. (Zenger) Soit x1, . . . , xn des vecteurs lineaires independants de l’espace de Hilbert

complexe H et α1, . . . , αn ∈ R∗+ tel que

n∑j=1

αj = 1. Alors il existe w ∈ Spanx1, . . . , xn,

w =n∑

j=1

βjxj tels que ‖w‖ = 1 et < βjxj , w >= αj, (j = 1, n).

Corollaire 2.4. Soit T = (T1, . . . , Tm) ∈ (L(H))m un systeme commutatif, absolument continu

et F ⊂ σp(T ) (ou F ⊂ σp(T ∗)) un ensemble fini. Pour tout αλλ∈F ⊂ C, il existe u, v ∈ H, telque ||v||2 (resp. ||u||2) =

∑λ∈F

|αλ| et∑

λ∈F

αλEλ = u ⋄T v

De plus, si αλλ∈F ⊂ R+, alors on peut choisir v = u. En particulier, pour tout αλλ∈F ⊂ Cil existe u1, u2, u3, u4 dans H tels que:

∑

λ∈F

αλEλ = u1⋄Tu1 − u2⋄Tu2 + iu3⋄Tu3 − iu4⋄Tu4

et4∑

k=1

||uk||2 ≤√2∑

λ∈F

|αλ|.

Preuve: Pour λ ∈ F soit xλ ∈ Ker(T −λ)\0. (resp. xλ ∈ Ker(T ∗−λ)\0). En appliquantle resultat anterieur du systeme lineaire independant xλλ∈F il resulte qu’il existe v′ =

∑λ∈F

βλxλ

avec ||v′|| = 1 et

< βλxλ, v′ >=

|αλ|∑µ∈F

|αµ|, (λ ∈ F )

Si αλ = eiθλ |αλ| en notant u′ :=∑λ∈F

βλeiθλxλ on obtient:

< h, u′ ⋄T v′ >=< h(T )u′, v′ >=∑

λ∈F

βλeiθλ < h(T )xλ, v

′ >=

=∑

λ∈F

βλeiθλ < h(λ)xλ, v

′ >=∑

λ∈F

eiθλh(λ)|αλ|∑

µ∈F

|αµ|=

=1∑

µ∈F

|αµ|∑

λ∈F

αλh(λ) =1∑

µ∈F

|αµ|< h,

∑

λ∈F

αλEλ >, (h ∈ H∞(Dm) ).

Pour u := (∑µ∈F

|αµ|)12u′, respectivement v := (

∑µ∈F

|αµ|)12 v′ on obtient la premiere conclusion

ci-dessus. Pour le cas particulier on applique la conclusion obtenue pour les systemes de scalaires:


(Reαλ)+λ∈F , (Reαλ)

−λ∈F , (Imαλ)+λ∈F et respectivement (Imαλ)

−λ∈F (ou a+ (a−)note la partie positive (negative) du numero reel a)

Remarquons que, bien que la premiere identite de l’enonce semble une combinaison lineairefinie aux coefficients complexes d’elements de type Eλ ∈ Q (evaluations) on peut l’ecrire commeun element de la forme u ⋄T v ∈ Q, elle n’offre pas un controle sur la norme de vecteur u (resp. v).

Nous allons voir dans le dernier paragraphe comment on peut appliquer ces resultats dansl’etude des systemes commutatifs finis de contractions.

Pour T comme plus haut et k ∈ Nm (resp. k ∈ N), nous notons T k = T k11 . . . T km

m (resp.T k = T k

1 . . . T km). Une dilatation isometrique de T est un systeme commutatif d’isometries, V =

(V1, . . . , Vm) ∈ (L(K))m

defini sur un espace de Hilbert K ⊃ H tel que

T kh = PKHV kh, (h ∈ H, k ∈ Nm)

ou PKH est la projection orthogonale de K sur H. La dilatation V ∈ (L(K))

mest dite minimale si K

est le plus petit sous-espace invariant pour V qui contient H c’est-a-dire K =∨V kH : k ∈ Nm.

(voir [2] ou [11] Chap. 1)Nous allons noter [H, T ] un systeme commutatif T = (T1, . . . , Tm) ∈ (L(H))

m.

Si [K, V ] est une dilatation isometrique minimale de [H, T ] alors [K, V ∗] est une extensioncoisometrique minimale de [H, T ∗], et si [H, T ] est absolument continu alors [H, T ∗] et [K, V ] sontaussi (voir Cor.1.9 de [9]).

La proposition suivante (voir [9]) fournit pour les m−uplets d’isometries un analogue de ladecomposition de Wold d’une isometrie (rappelons que un sous espace S est dit reduisant pour lesysteme V s’il est invariant pour V et pour son adjoint V ∗):

Proposition 2.5. Soit [K, V = (V1, . . . , Vm)] un m−uplet commutatif d’isometries. Alors il existeune decomposition orthogonale unique

K = S ⊕Rtelles que:

i) S,R sont des sous-espaces reduisants pour V,

ii) V|S est completement non-unitaire, et V|R est unitaire .

De plus, S = x ∈ K : infn∈Nm

||V ∗nx|| = 0, et R est le plus grand sous-espace reduisant pour V

telle que la restriction de V a ce sous-espace soit unitaire.

Il sera utile de remarquer que dans le cas d’une seule isometrie l’espace S admet la description:

(3) S =∨

z∈D

Ker(V ∗ − z)

Il faut remarquer aussi que si le systeme T a la propriete K.,0 et [K, V ] est une d.i.m. de T (s’ilexiste) alors R = (0).

Le resultat suivant donne une structure de l’espace S de la decomposition anterieure qui serautilisee dans la suite pour la demonstration des certaines conditions d’annulations (les conditions2.2 ou 2.3 de Def. 2.1).


Proposition 2.6. Pour tout m−uplet commutatif d’isometries [K, V ] il existe une decompositionorthogonale

(4) K = S1 ⊕ . . .⊕ Sm ⊕ Sm+1

telle que, pour tout k ∈ 1, . . . ,m on a:

i) Rk := Sk+1 ⊕ . . .⊕ Sm ⊕ Sm+1 est invariant pour Vj , (j = 1,m),

ii) Rk−1 := Sk ⊕Rk est la decomposition Wold de Vk|Rk−1R0 := K)

iii) Rk est reduisant pour Vj , et Vj |Rkest unitaire, (j = 1, k).

Preuve: C’est une demonstration par induction qui essentiellement utilise la decompositionde Wold pour une seule isometrie et le fait que l’espace R de cette decomposition est invariantpour tout operateur qui commute avec cette isometrie.

Remarqons que, par rapport a la decomposition orthogonale (4), l’isometrie Vj admet larepresentation matricielle:

(5) Vj =

X1,1 0 . . . 0 0 . . . 0...

......

......

Xj,1 Xj,2 . . . Xj,j 0 . . . 00 0 . . . 0 Xj+1,j+1 . . . 0...

......

......

0 0 . . . 0 0 . . . Xm+1,m+1

ou Xi,k : Sk → Si, Xi,k := PSiVjx, (x ∈ Sk)

Le resultat de la proposition anterieure a ete utilise par B. Chevreau pour demontrer le theoremesuivant, resultat qui generalise dans le cas de systemes le resultat de G. Robel (voir [13]).

Theoreme 2.7. (B. Chevreau, 1998) Fie T = (T1, . . . , Tm) ∈ (L(H))m

un systeme commutatif decontractions, absolument continu et de classe A. Si T admet une dilatation isometrique, alors,pour tout λ ∈ σH

e (T ) ∩Dm nous avons que Eλ est dans l’ensemble X0(T ).

Aussi, en utilisand la meme decomposition, B. Chevreau a caracterisee la propriete (Aℵ0)

pour un systeme commutatif de contractions, absolument continu et qui admette une dilatationisometrique, avec la propriete X0,1. Plus precisement:

Theoreme 2.8. (B. Chevreau, 1998) Soit T = (T1, . . . , Tm) ∈ (L(H))m

un m−uplet absolumentcontinu, et qui admet une dilatation isometrique. Les affirmations suivantes sont equivalentes:

i) Il existe 0 ≤ θ < γ tel que T a la propriete Xθ,γ ;

ii) T a la propriete X0,1;

iii) T a la propriete (Aℵ0);

iv) Il existe n ≥ 1 tel que T ⊕ . . .⊕ T︸︷︷︸n fois

a la propriete (Aℵ0).


Dans le cas d’une seule isometrie de classe A on sait que si une suite (xn)n≥0 ⊂ S, (l’espacedonnee par la proposition 2.5) est faiblement convergente vers zero alors lim

n→∞||xn ⋄V w|| = 0, pour

tout w ∈ K. Dans le cas d’un systeme commutatif avec plus de deux isometries le resultat n’estplus vrai (sous ces hypotheses). Par exemple, si Q := Z2 \ (−Z2

+), considerant l’espace de Hilbert

ℓ2(Q,C) :=

(xm,n)(m,n)∈Q :

∑(m,n)∈Q

|xm,n|2 < ∞

et les isometries commutatives:

V,W : ℓ2(Q,C) → ℓ2(Q,C), V (ei,j) = ei+1,j , W (ei,j) = ei,j+1, (i, j) ∈ Q

ou

ei,j ∈ ℓ2(Q,C), ei,j(m,n) =

1 , (i, j) = (m,n)0 , (i, j) 6= (m,n)

,

nous aurons que la paire d’isometries (V,W ) est de classe A. Considerant la suite orthogonale(xn)n≥1 ⊂ ℓ2(Q,C) ou xn := e1,−n, (n ≥ 1), nous aurons que lim

m→∞||V ∗mxn|| = 0, (n ≥ 1), (et

allors en ∈ S) maislim

n→∞< Wnxn, e1,0 >= 1 6= 1 et donc w ∈ ℓ2(Q,C) : lim

n→∞||xn ⋄(V,W ) w|| = 0 6= ℓ2(Q,C). On

peut expliquer tout cela par le fait que limm→∞

||W ∗mxn|| = limm→∞

||e1,−n−m|| = 1.

Nous avons cependant le resultat d’annulation a l’infini suivant analogue au cas d’une seuleisometrie. La demonstration est semblable a celle du theoreme 2.7.

Lemme 2.9. Soit [K, V = (V1, . . . , Vm)] un systeme commutatif d’isometries, absolument continuet de classe A. Alors, pour toute suite faiblement convergente vers zero (xn)n≥0 ⊂ K ayant lapropriete lim

n→∞||V ∗

jnxp|| = 0, (p ≥ 0, j = 1,m), nous avons que

(6) limn→∞

||xn ⋄V w|| = 0, (w ∈ K).

Preuve: Nous allons faire la demonstration par induction sur m ≥ 1. Soit

Um := w ∈ K : limn→∞

||xn ⋄V w|| = 0

Il s’agit de montrer que Um = K. Pour m = 1 soit

K = S1 ⊕R1

la decomposition Wold de l’espace K relative a l’isometrie V1. Nous avons xn ∈ S1, (n ≥ 0), etS1 et R1 sont reduisants pour V1. Il suffit de montrer que S1, R1 ⊂ U1. Si w ∈ R1, commeh(V1)xn ∈ S1, (h ∈ H∞(D)) il resulte que xn ⋄V w = 0, (n ≥ 0) et donc w ∈ U1.

Comme S1 =∨t∈D

Ker (V ∗1 − t), pour l’inclusion S1 ⊂ U1 il suffit de montrer que Ker (V ∗

1 − t) ⊂U1 pour chaque t ∈ D. Soit donc t ∈ D et w ∈ Ker (V ∗

1 − t). Soit gn ∈ H∞(D) tel que ||gn||∞ = 1et ||xn ⋄V w|| = =< gn(V1)xn, w >. Pour un ∈ H∞(D) tel que

gn(z) = gn(t) + (z − t)un(z), (z ∈ D)

Il resulte que:

limn→∞

||xn ⋄V w|| = limn→∞

< gn(V1)xn, w >

= limn→∞

gn(t) < xn, w >= 0


Soit maintenant m ≥ 2, [K, V = (V1, . . . , Vm)] et (xn)n≥0 ⊂ K donnes avec les proprietes del’enonce. Conformement a la proposition 2.6, nous avons la decomposition orthogonale:

K = S1 ⊕ . . .⊕ Sm ⊕ Sm+1

ou Sm+1 est reduisant pour V1, . . . , Vm, et Vj |Sm+1, (j = 1,m) sont unitaires. En particulier il

resulte que xn ∈ S⊥m+1. D’une maniere analogue a celle de plus haut il resulte que Sm+1 ⊂ Um.

Nous allons montrer par induction que Sk ⊂ Um, pour tout k = 1,m.Soit w ∈ S1 =

∨t∈D

Ker (V ∗1 − t). On peut supposer que V ∗

1 w = tw, pour un certain t ∈ D. Il

existe gn, un ∈ H∞(Dm) tels que ||gn||∞ = 1, ||xn ⋄V w|| =< gn(V )xn, w > et

gn(z) = gn(t, z2, . . . , zm) + (z1 − t)un(z), (z ∈ Dm)

Notant fn(z) := gn(t, z), (z ∈ Dm−1) nous aurons que ||fn||∞ ≤ 1 et

< gn(V )xn, w >=< fn(V2, . . . , Vm)xn, w >≤ ||xn ⋄V (1)

w||ou V (1) := (V2, . . . , Vm). Comme, conformement a l’hypothese d’induction le deuxieme membre dela derniere egalite converge vers zero pour n tendant vers l’infini, il resulte S1 ⊂ Um.

Soit maintenant 1 ≤ k < m et supposons, comme hypothese d’induction, que Sj ⊂ Um pourj = 1, k. Nous allons etablir w ∈ Sk+1. Comme Sk+1 =

∨t∈D

Ker ((Vk+1|Rk)∗ − t) il suffit de

supposer que (Vk+1|Rk)∗w = tw pour un certain t ∈ D. Soit gn, un ∈ H∞(Dm) tel que ||gn||∞ = 1,

||xn ⋄V w|| = < gn(V1)xn, w > et

(7) gn(z) = gn(z1, . . . , zk, t, zk+2, . . . , zm) + (zk+1 − t)un(z), (z ∈ Dm)

Nous aurons

||un||∞ ≤ 2

1− |t| ||gn||∞, (n ∈ N)

Notant fn(z1, . . . , zk, zk+2, . . . , zm) := gn(z1, . . . , zk, t, zk+2, . . . , zm), nousobtenons la suite bornee (fn)n≥0 ⊂ H∞(Dm−1). De (7) il resulte que:

(8) < gn(V )xn, w >=< fn(V(k+1))xn, w > + < un(V )xn, (V

∗k+1 − t)w >

ou V (k+1) := (V1, . . . , Vk, Vk+2, . . . , Vm). Conformement a l’hypothese d’induction correspondantau (m− 1)−uplet V (k+1) et a la suite (xn)n≥0 nous avons que:

(9) limn→∞

< fn(V(k+1))xn, w >= 0

(car | < fn(V(k+1))xn, w > | ≤ ||xn ⋄V (k+1)

w||). Comme

(V ∗k+1 − t)w = PRk

(V ∗k+1 − t)w + PS1⊕...⊕Sk

(V ∗k+1 − t)w =

((Vk+1|Rk

)∗ − t)w + PS1⊕...⊕Sk

(V ∗k+1 − t)w = PS1⊕...⊕Sk

(V ∗k+1 − t)w

il resulte, conformement a l’hypothese d’induction de cette etape et du fait que la suite (un)n≥0

est bornee, que


limn→∞

< un(V )xn, (V∗k+1 − t)w >= lim

n→∞< un(V )xn, PS1⊕...⊕Sk

(V ∗k+1 − t)w >= 0.

Les relations (7)− (9) montrent que limn→∞

< gn(V )xn, w >= 0, donc Sk+1 ⊂ Um, ce qui acheve la

demonstration du lemme.

3 L’ensemble d’approximation Zθ.

Nous allons introduire maintenant un nouvel ensemble d’approximation qui permet d’obtenir, souscertaines hypotheses, la propriete (A1/8) pour un m−uplet de contractions. Mais d’abord il fautfixer le cadre:

Dans ce qui suit, T ∈ (L(H))m

sera un m−uplet commutatif de contractions, de classe A, quiadmet une dilatation isometrique. Nous fixons des extensions coisometriques minimales [K, C], et[K∗, C∗] de [H, T ], respectivement [H, T ∗] et soit

K = S ⊕R, C = S∗ ⊕R

etK∗ = S∗ ⊕R∗, C = S∗

∗ ⊕R∗

leurs decompositions de type Wold donnees par la Proposition 2.6. Nous notons Q, Q∗, A et A∗les projections orthogonales sur les espaces S, S∗, R et respectivement R∗. Pour p ∈ [1,∞] et µune mesure borelienne sur Tm (ou T := z ∈ C : |z| = 1) on note:

Hp(µ) := C[z1, . . . , zm]Lp(µ)

et

Q(µ) := L1(µ)/⊥H∞(µ)

le predual de H∞(µ) (donc (Q(µ))∗ = H∞(µ)).En [9] J. Eschmeier montre le resultat suivant:

Lemme 3.1. (Lemme 2.4 de [9]) Avec les notations ci-dessus, supposons que R 6= (0). Alors ilexiste µ une mesure Henkin positive sur Bor(Tm) avec µ(Tm) = 1 et il existe un sous-espace R0

de R, reduisant pour R tels que:i) R0 := R|R0

est unitairement equivalent a Mz sur L2(µ);

ii) Le sous-espace R+0 de R0 qui corresponde a H2(µ) par l’equivalence unitaire de i)

verifie R+0 ⊂ AH.

Si E est la mesure spectrale de R et x, y ∈ R alors x ·y ∈ L1(µ) note la derivee Radon-Nikodymde la mesure

µx,y : Bor(Tm) → C, µx,y(A) =< E(A ∩ σ(T ))x, y >

par rapport a µ. On note x⊙µ y := [x · y] ∈ Q(µ). Soit

r∗ : Q(µ) → Qle preadjoint de l’homomorphisme contractif f∗−continu qui prolonge l’application identite surA(D)

m:


r : H∞(Dm) → H∞(µ)

(voir [9]). Il est facile de voir que r∗(x⊙µ y) = x ⋄C y, pour tout x, y ∈ R.Nous notons µ∗ la mesure donnee par le lemme anterieur pour [R∗, R∗].

Definition 3.2. Pour θ ∈ [0, 1] on definit l’ensemble Zθ(T ) comme l’ensemble des elements [f ]dans le predual Q de H∞(Dm), pour lesquelles il existe des suites de vecteurs (xn), (bn) ⊂ H,(yn) ⊂ K, et (an) ⊂ K∗ dans la boule unite fermee de ces espaces, telles que:

(z1) limn→∞

||[f ]− (xn ⋄C yn + an ⋄C∗∗bn)|| ≤ θ,

(z2) limn→∞

||xn ⋄T w|| = limn→∞

||w ⋄T bn|| = 0, (w ∈ H),

(z3) limn→∞

||w ⋄C yn|| = 0, (w ∈ S),

(z4) limn→∞

||an ⋄C∗∗w|| = 0, (w ∈ S∗).

Pour 0 ≤ θ < γ ≤ 1 on dira que T a la propriete Zθ,γ si l’enveloppe absolument convexe fermee deZθ(T ) contient la boule fermee de Q de rayon γ.

Le lemme suivant, dont la demonstration suit les memes idees que celles de la Proposition 3.2,de [9], permet d’ameliorer des approximations successives de longueur deux pour les elements deQ.

Lemme 3.3. Supposons que T a la propriete Zθ,γ pour certains 0 < θ < γ ≤ 1. Alors, pour tous[f ] ∈ Q, 0 < ρ < 1, δ > 0, x, b ∈ H, y ∈ S, w ∈ R, a ∈ S∗, et c ∈ R∗, tels que

||[f ]−(x ⋄C (y + w) + (a+ c) ⋄C∗

∗b)|| < δ,

il existe x′, b′ ∈ H, y′ ∈ S, w′ ∈ R, a′ ∈ S∗, et c′ ∈ R∗ tels que:

||[f ]−(x′ ⋄C (y′ + w′) + (a′ + c′) ⋄C∗

∗b′)|| < θ

γδ,

avec ||x′−x||, ||b′−b|| ≤ 3(

δγ

)1/2, ||y′−y||, ||a′−a|| ≤

(δγ

)1/2et ||w′|| < 1

ρ

(||w||+

(δγ

) 12

), ||c′|| <

1ρ

(||c||+

(δγ

) 12

).

Preuve: Soit d := ||[f1]|| ou [f1] := [f ]− x ⋄C (y+w)+ +(a+ c) ⋄C∗∗b. On peut supposer que

d 6= 0. Soit ε > 0 tels que dγ (θ + ε) + 7ε < θ

γ δ et dγ + ε < δ

γ . Comme ||γd [f1]|| = γ il resulte qu’il

existe N ∈ N, (Zj)j=1,N ⊂ Zθ(T ) et α1, . . . , αN ∈ C tels que:

||[f1]−N∑

j=1

αjZj || ≤ ε,N∑

j=1

|αj | ≤d

γ

Soit η > 0 fixe tel que N(N + 1)η < ε . Comme Zj ∈ Zθ(T ) il resulte que, pour tout j = 1, N , ilexiste xj , bj ⊂ (H)1, yj ⊂ (K)1 et bj ∈ (K∗)1 tels que:

||Zj − (xj ⋄C yj + aj ⋄C∗∗bj)|| < θ + ε,


||x ⋄C Qyj = ||Qx ⋄C yj || < η, ||xj ⋄C y|| = ||xj ⋄T PHy|| < η,

||xj ⋄C Qyk|| = ||xj ⋄T PHQyk|| = ||Qxj ⋄C yk|| < η, (j, k = 1, N, j 6= k)

||xj ⋄C w|| = ||xj ⋄T PHw|| < η

| < xj , xk > |, | < yj , Qyk > | = | < Qyj , yk > | < η

||a ⋄C∗∗bj || = ||bj ⋄C∗ a|| = ||bj ⋄T

∗PHa|| = ||PHa ⋄T bj || < η,

||Q∗aj ⋄C∗∗bk|| = ||PHQ∗aj ⋄T bk|| = ||aj ⋄C∗

∗Q∗bk|| < η, (j, k = 1, N, j 6= k)

||PHc ⋄T bj ||, ||Q∗aj ⋄C∗∗b|| = ||aj ⋄C∗

∗b|| < η,

| < aj , Qak > |, | < bj , bk > | < η, (j, k = 1, N, j 6= k).

Nous notons:

QN := x ⋄C y +

N∑

j=1

αjxj ⋄C Qyj , AN := x ⋄C w +

N∑

j=1

αjxj ⋄C Ayj ,

Q∗,N := a ⋄C∗∗b+

N∑

j=1

αjQ∗aj ⋄C∗∗bj , A∗ := c ⋄C∗

∗b+

N∑

j=1

αjA∗aj ⋄C∗∗bj .

Soit βj ∈ C tel que β2j = αj , (j = 1, N). Si on note

x′1 :=

N∑

j=1

βjxj ∈ H, y′1 :=N∑

j=1

βjQyj ∈ S

a′1 :=N∑

j=1

βjQ∗aj ∈ S∗, b′1 :=N∑

j=1

βjbj ∈ H,

Alors

||((x+ x′

1) ⋄C (y + y′1))−QN || ≤

≤N∑

j=1

|βj | ||x ⋄C Qyj ||+N∑

j=1

|βj | ||xj ⋄C y||+N∑

j, k = 1,j 6= k

|βj | |βk| ||xj ⋄C Qyk|| < ε

De la meme maniere on obtient:

||((a+ a′1) ⋄C∗

∗(b+ b′1)

)−Q∗,N || < ε

De plus, nous avons que:


||a′1||2 =N∑

j=1

|αj | ||Q∗aj ||2 +N∑

j, k = 1,j 6= k

βjβk < aj , Q∗ak >≤ d

γ+N(N + 1)η <

δ

γ

et de facon analogue

||b′1||2, ||x′1||2, ||y′1||2 ≤ δ

γ

Comme

||A(x+ x′1)⊙µ w +

N∑

j=1

αjAxj ⊙µ Ayj −A(x+ x′1)⊙µ w|| ≤ d

γ<

δ

γ

conformement a la Proposition 2.6 de [9], il resulte qu’il existe x′2 ∈ H et w′

2 ∈ R tels que:

||A(x+ x′1)⊙µ w +

N∑

j=1

αjAxj ⊙µ Ayj −A(x+ x′1 + x′

2)⊙µ w′|| < ε

2,

||x′2|| ≤ 2

(δ

γ

)1/2

, ||Qx′2|| ≤

ε

||y + y′1||+ 1

et

||w′|| ≤ 1

ρ

(||w||+

(δ

γ

)1/2)

Comme

||Ax′1 ⋄C w|| ≤

N∑

j=1

|βj | ||xj ⋄T PHw|| ≤ Nη <ε

2

etAN −A(x+ x′

1 + x′2) ⋄C w′ = −Ax′

1 ⋄C w+

+r∗

A(x+ x′

1)⊙µ w +

N∑

j=1

αjAxj ⊙µ Ayj −A(x+ x′1 + x′

2)⊙µ w′

,

il resulte que

||AN −A(x+ x′1 + x′

2) ⋄C w′|| < ε.

En utilisant encore une fois la Proposition 2.3 de [9] mais par rapport aux Q(µ∗) et R∗ etcomme

||A∗(b+ b′1)⊙µ∗ c+N∑

j=1

αjA∗bj ⊙µ∗ aj −A∗(b+ b′1)⊙µ∗ c|| ≤ d

γ<

δ

γ

on obtient qu’il existe b′2 ∈ H et c′ ∈ R∗ tels que:


||A∗(b+ b′1)⊙µ∗ c+N∑

j=1

αjA∗bj ⊙µ∗ A∗aj −A(b+ b′1 + b′2)⊙µ∗ c′|| < ε

2,

||b′2|| ≤ 2

(δ

γ

)1/2

, ||Qb′2|| ≤ε

||b+ b′1||+ 1

et

||c′|| ≤ 1

ρ

(||c||+

(δ

γ

)1/2)

Alors,

||A∗,N−c′ ⋄C∗∗A(b+b′1 + b′2)||= ||A∗b ⋄C∗ c+

N∑

j=1

αjA∗bj ⋄C∗ aj −A(b+ b′1 + b′2) ⋄C∗ c′|| ≤

≤ ||A∗(b+ b′1)⊙µ∗ c+

N∑

j=1

αjA∗bj ⊙µ∗ A∗aj −A(b+ b′1 + b′2)⊙µ∗ c′||+ ||c ⋄C∗∗b′1|| < ε

(parce que ||c ⋄C∗∗b′1|| ≤

N∑j=1

||PHc ⋄C∗∗bj || ≤ Nη < ε/2).

Mais

||[f ]−((x+ x′

1 + x′2) ⋄C (y+ y′1 + w′) + (a+a′1 + c′) ⋄C∗

∗(b+ b′1 + b′2)

)||

≤ ||[f ]−N∑

j=1

αjZj ||+N∑

j=1

|αj |||Zj − (xj ⋄T yj + aj ⋄T bj)||+

+ ||QN − (x+ x′1) ⋄C (y + y′1)||+ ||Q∗,N − (a+ a′1) ⋄C∗

∗(b+ b′1)||+

+ ||AN −A(x+ x′1 + x′

2) ⋄C w′||+ ||A∗,N − c′ ⋄C∗∗A∗(b+ b′1 + b′2)||+

+ ||x′2|| ||y + y′1||+ ||b′2|| ||a+ a′1|| ≤

d

γ(θ + ε) + 7ε <

θ

γδ.

Pour x′ := x+ x′1 + x′

2, b′ := b+ b′1 + b′2, y

′ := y + y′1 et a′ = a+ a′1 on obtient les affirmationsde l’enonce.

Nous allons voir comme on peut obtenir un factorisation exacte en utilisant la methode d’approximationssuccessives qui se base sur le dernier lemme. La demonstration est similaire a celle du corollaire3.3 de [9].

Corollaire 3.4. Supposons que T a la propriete Zθ,γ pour certains 0 ≤ θ < γ ≤ 1. Alors, pourtous [f ] ∈ Q, 0 < ρ < 1, δ > 0, x, b ∈ H, y ∈ S, w ∈ R, a ∈ S∗, et c ∈ R∗, tel que

||[f ]−(x ⋄C (y + w) + (a+ c) ⋄C∗

∗b)|| < δ,

il existe x′, b′ ∈ H, y′ ∈ S, w′ ∈ R, a′ ∈ S∗, et c′ ∈ R∗ tels que:

[f ] = x′ ⋄C (y′ + w′) + (a′ + c′) ⋄C∗∗b′


||x′ − x||, ||b′ − b|| ≤ 3αδ1/2,

||y′ − y||, ||a′ − a|| ≤ αδ1/2,

||w′|| < 1

ρ

(||w||+ δ

12

), ||c′|| < 1

ρ

(||c||+ δ

12

).

ou α := 1/(γ1/2 − θ1/2).

Preuve: On peut supposer que θ > 0. Soit (sj) une suite decroissante avec s0 = 1 et limn→∞

sn =

ρ. Pour n ≥ 0 soit ρn := sn/sn−1. Par l’application iterative du lemme 3.3 nous obtenons lessuites (xn), (bn) ⊂ H, (yn) ⊂ S, (wn) ⊂ R, (an) ⊂ S∗, (cn) ⊂ R∗ tels que:

||[f ]−(xn ⋄C (yn + wn) + (an + cn) ⋄C∗

∗bn

)|| <

(θ

γ

)n

δ

||xn − xn−1||, ||bn − bn−1|| ≤ 3

(δ

γ

)1/2(θ

γ

)n−12

,

||yn − yn−1||, ||an − an−1|| ≤(δ

γ

)1/2(θ

γ

)n−12

,

||wn|| <1

ρn

(||wn−1||+

(δ

γ

) 12(θ

γ

)n−12

),

||cn|| <1

ρn

(||cn−1||+

(δ

γ

) 12(θ

γ

)n−12

).

pour tout n ≥ 0 (ou x0 = x, . . . , b0 = b).Les inegalites precedentes montrent que les suites (xn), (bn), (yn) et (an) sont des suites de

Cauchy et que leurs limites, x′, b′, y′, a′, satisfont les inegalites de l’enonce.Comme

sn||wn|| < sn−1

(||wn−1||+

(δ

γ

) 12(θ

γ

)n−12

)

il resulte que

||wn|| <1

ρ

(||w||+ αδ1/2

)

et une inegalite pareille pour cn. Ainsi, quitte a passer a des sous-suites, nous pouvons supposerqu’elles convergent faiblement vers w′ respectivement c′. Avec ces vecteurs sont verifiees toutes lesconclusions de l’enonce.


4 Multi-contractiona spectre de Harte dominant.

Nous allons montrer que tout m−uplet de contractions commutatives, absolument continu, ayantle spectre de Harte dominant dans Dm a la propriete (A1/8).

Rappelons qu’un sous-espace M ⊂ H est dit semi-invariant pour le systeme T s’il existe deuxsous-espaces invariant pour T , M1 ⊂ M2, tels que M = M2 ⊖M1 (c’est-a-dire M2 = M⊕M1).Si M est semi-invariant pour T , alors on va noter TM le systeme TM := (T1,M, . . . , Tm,M) ∈(L(M))

m, ou Tj,Mx := PMTjx, x ∈ M (la compression de Tj a M.)

Il sera utile de montrer d’abord un lemme technique:

Lemme 4.1. Soit T ∈ (L(H))m

un m−uplet commutatif de contractions, dans la classe A. SiM et N sont deux sous-espaces fermes de H, semi-invariants pour T respectivement T ∗ et decodimension finie de sorte que

(σHe (T ) ∪ Λ(TM) ∪ Λ(T ∗N )

)∩Dm

est dominant dans Dm alors, pour tout [f ] ∈ Q, ε > 0 et W ⊂ H un sous-ensemble fini, il existedes vecteurs x, y, uk, vk ∈ H, (k = 1, 4) tels que:

i) ||[f ]−(x ⋄T y +

4∑k=1

ikuk ⋄T uk +4∑

k=1

ikvk ⋄T vk

)|| ≤ ε;

ii) ||a||2 +4∑

k=1

||uk||2 +4∑

k=1

||vk||2 ≤√2||[f ]||, (a ∈ x, y);

iii) ||x ⋄T w||+ ||w ⋄T y|| ≤ ε, (w ∈ W ∪ uk, vk : k = 1, 4);iv) lim

p→∞||T p

j uk|| = limp→∞

||(T ∗j )

pvk|| = 0, (k = 1, 4, j = 1,m).

Preuve: Soit [f ], ε et W comme dans l’enonce. On peut supposer que ||[f ]|| = 1.Du fait que

(σHe (T ) ∪ Λ(TM) ∪ Λ(T ∗N )

)∩Dm

soit dominant dans Dm, il existe trois collections finies Je ⊂ σHe (T ), Jp ⊂ Λ(TM) et J∗

p ⊂ Λ(T ∗N )et un ensemble fini (αλ)λ∈Je∪Jp∪J∗

p⊂ C telles que:

(10) ||[f ]−

∑

λ∈Je

αλEλ +∑

λ∈Jp

αλEλ +∑

λ∈J∗p

αλEλ

|| ≤ ε

2,

∑

λ∈Je∪Jp∪J∗p

|αλ| ≤ 1.

Pour factoriser la deuxieme et la troisieme somme ci-dessus, on peut appliquer maintenantle corollaire 2.4 pour TM, respectivement pour T ∗N . On en deduit qu’il existe les vecteursu1, . . . , u4 ∈ ∨Ker (TM − λ) : λ ∈ Jp et v1, . . . , v4 ∈ ∨Ker (T ∗N − λ) : λ ∈ J∗

p telsque:

(11)∑

λ∈Jp

αλEλ =

4∑

k=1

ikuk ⋄TM uk,∑

λ∈J∗p

αλEλ =

4∑

k=1

ikvk ⋄TN vk

avec


(12)4∑

k=1

||uk||2 ≤√2∑

λ∈Jp

|αλ|,4∑

k=1

||vk||2 ≤√2∑

λ∈J∗p

|αλ|.

Conformement du theoreme 2.7, la somme∑

λ∈Je

αλEλ est dans X0 et, donc, il existe x, y ∈ Htels que:

(13) ||∑

λ∈Je

αλEλ − x ⋄T y|| ≤ ε

2, ||x||2, ||y||2 ≤

∑

λ∈Je

|αλ|;

et||x ⋄T w||+ ||w ⋄T y|| ≤ ε, (w ∈ W ∪ uk, vk : k = 1, 4)

Les relations (10) - (13) et le fait que x⋄TV y = x⋄T y, pour tout x, y ∈ V, si V est semi-invariantpour T , conduisent aux conclusions i), ii) et iii).

Il a reste demontrer la derniere conclusion. Soit j ∈ 1,m et k = 1, 4 fixes. Parce que uk ∈∨λ∈Jp

Ker (TM−λ), il resulte que limp→∞

||PMT pj uk|| = lim

p→∞||(Tj,M)

puk|| = 0. Mais T est absolument

continu et alors la suite (T pj uk)p converge faiblement vers zero, donc (PM⊥T p

j uk)p aussi. Comme

dimM⊥ < ∞ il resulte que la derniere suite converge fortement vers zero et donc limp→∞

||T pj uk|| = 0.

Dans la meme maniere il resulte que limp→∞

||T ∗jpvk|| = 0 ce qui finit la demonstration.

Nous allons voir comment appliquer ce lemme dans un processus inductif de construction defactorisations approximatives. Il sera utile, pour simplifier l’ecriture, d’introduire un nouvelle no-tation: pour un entier n ≥ 1 et un m−uplet T = (T1, . . . , Tm) ∈ (L(H))

mon note T (n) le m−uplet

(T(n)1 , . . . , T

(n)m ) ∈

(L(H(n))

)m, ou pour un operateur S ∈ L(H) on note S(n) l’ampliation d’ordre

n de S, c’est-a-dire, l’operateur S(n) : H(n) := H× . . .×H︸︷︷︸n fois

→ H(n), defini par S(n)(x1, . . . , xn)) =

(Sx1, . . . , Sxn). Si T est absolument continu (resp. de classe A) alors T (n) l’est aussi. De plus,

pour x = (x1, . . . , xn) et y = (y1, . . . , yn) dans H(n) on a x ⋄T (n)

y = x1 ⋄T y1 + . . .+ xn ⋄T yn.

Proposition 4.2. Soit T ∈ (L(H))m

un m−uplet commutatif de contractions, absolument con-tinu, qui admet une dilatation isometrique. Supposons que l’ensemble σH(T ) ∩Dm est dominantdans Dm. Alors, pour tout [f ] ∈ Q il existe des suites (un), (u

′n), (vn), (v

′n) ⊂ H(4) faiblement

convergentes vers zero tels que:i) ||un||2 + ||vn||2, ||u′

n||2 + ||v′n||2 ≤√2||[f ]||;

ii) limn→∞

||vn ⋄T (4)

w|| = limn→∞

||w ⋄T (4)

u′n|| = 0, (w ∈ H(4))

iii) limn→∞

||[f ]− (un ⋄T (4)

u′n + vn ⋄T (4)

v′n)|| = 0.

Preuve: Le fait que σH(T )∩Dm est dominant dans Dm assure l’appartenance de T a la classeA.

Soit [f ] dans Q. Nous choisissons aussi une suite de vecteurs (wn) dense dans H et une suitede nombres reels positifs (εn) decroissante vers 0.

En appliquant pour [f ] le lemme 4.1 avec M = N = H, W = w0 et ε0 > 0 il resultel’existence des vecteurs x0, y0, uk,0, vk,0 ∈ H, (k = 1, 4) tels que:

||[f ]−(x0 ⋄T y0 +

4∑

k=1

ikuk,0 ⋄T uk,0 +

4∑

k=1

ikvk,0 ⋄T vk,0

)|| ≤ ε0;


||a0||2 +4∑

k=1

||uk,0||2 +4∑

k=1

||vk,0||2 ≤√2||[f ]||, (a0 ∈ x0, y0);

||x0 ⋄T w||+ ||w ⋄T y0|| ≤ ε0, (w ∈ W ∪ uk,0, vk,0 : k = 1, 4);

limp→∞

||T pj uk,0|| = lim

p→∞||(T ∗

j )pvk,0|| = 0, (k = 1, 4, j = 1,m).

Soit J0e , J

0p et J∗,0

p les collections utilisees dans la demonstration du lemme 4.1. On pose

M⊥1 :=

∨

λ∈J0p

Ker (T − λ), N⊥1 :=

∨

λ∈J∗,0p

Ker (T ∗ − λ)

Alors M1 et N1 sont semi-invariants pour T et en plus

Λ(T ) \ J0p ⊂ Λ(TM1

), Λ(T ∗) \ J∗,0p ⊂ Λ(T ∗

N1)

Comme (σHe (T )∩Dm)∪

(Λ(T ) \ J0

p

)∪(Λ(T ∗) \ J∗,0

p

)reste encore dominant dans Dm il resulte

que

(σHe (T ) ∪ Λ(TM1

) ∪ ˜Λ(T ∗N1))∩Dm

est dominant dans Dm. En appliquant encore une fois le lemme 4.1, pour le meme [f ], mais cettefois-ci avec M = M1, N = N1, ε = ε1 et W = w0, w1 on obtient des vecteurs x1, y1, uk,1 ∈M1, vk,1 ∈ N1, (k = 1, 4) tels que:

||[f ]−(x1 ⋄T y1 +

4∑

k=1

ikuk,1 ⋄T uk,1 +

4∑

k=1

ikvk,1 ⋄T vk,1

)|| ≤ ε1;

||a1||2 +4∑

k=1

||uk,1||2 +4∑

k=1

||vk,1||2 ≤√2||[f ]||, (a1 ∈ x1, y1);

||x1 ⋄T w||+ ||w ⋄T y1|| ≤ ε1, (w ∈ W ∪ uk,1, vk,1 : k = 1, 4);

limp→∞

||T pj uk,1|| = lim

p→∞||(T ∗

j )pvk,1|| = 0, (k = 1, 4, j = 1,m).

Soit J1e , J

1p et J∗,1

p les collectiones utilisees dans la demonstration du lemme 4.1 pour ces nou-velles donnees. On pose

M⊥2 := M⊥

1 ⊕∨

λ∈J1p

Ker (TM1− λ), N⊥

2 := N⊥1 ⊕

∨

λ∈J∗,1p

Ker (T ∗N1

− λ)

(M⊥2 (resp. N⊥

2 ) est invariant pour T (resp. T ∗)) et on reprend le processus pour M2 et N2 etainsi de suite.

Nous obtenons ainsi les suites orthogonales (et donc qui convergent faiblement vers zero)(xn), (yn), (uk,n), (vk,n) ⊂ H (k = 1, 4) telles que:


||[f ]−(xn ⋄T yn +

4∑

k=1

ikuk,n ⋄T uk,n +

4∑

k=1

ikvk,n ⋄T vk,n

)|| ≤ εn;

||an||2 +4∑

k=1

||uk,n||2 +4∑

k=1

||vk,n||2 ≤√2||[f ]||, (an ∈ xn, yn);

||xn ⋄T w||+ ||w ⋄T yn|| ≤ εn, (w ∈ w0, . . . wn ∪ uk,n, vk,n : k = 1, 4);

limp→∞

||T pj uk,n|| = lim

p→∞||(T ∗

j )pvk,n|| = 0, (k = 1, 4, j = 1,m).

Si un := (xn+u1,n,−u2,n, iu3,n,−iu4,n), u′n := (yn+u1,n, u2,n, u3,n, u4,n), vn := (v1,n,−v2,n, iv3,n,−iv4,n)

et v′n := (v1,n, v2,n, v3,n, v4,n) on obtient les conclusions i) et iii).La densite de la suite (wn) implique que lim

n→∞||xn ⋄T w|| = lim

n→∞||w ⋄T yn|| = 0. Comme

limp→∞

||(T ∗j )

pvk,n|| = 0, conformement au lemme 2.9 il resulte que lim

n→∞||vk,n ⋄T w|| = 0. De la

meme maniere on obtient que limn→∞

||w ⋄T un,k|| = 0, (w ∈ H), d’ou il resulte et la conclusion ii).

Theoreme 4.3. Soit T ∈ (L(H))m

un m−uplet commutatif de contractions, absolument continu,qui admet une dilatation isometrique et pour lequelle σH(T ) ∩Dm est dominant dans Dm. AlorsT (4) a la propriete Z0, 1√

2et donc (A1/2) aussi (ou, equivalente, T a la propriete (A1/8)). De plus,

si T est de type K0,0 alors T a la propriete (Aℵ0).

Preuve: Soit (wn) ⊂ H(4), (zn) ⊂ K(4), (z∗,n) ⊂ K(4)∗ denses dans leurs espaces, et (εn) ⊂ R∗

+

une suite decroissante vers zero. Si Ln ∈ Q, (n ≥ 1) sont definis par:

Ln(h) =1

((2n)!)m(∂2n1 . . . ∂2n

m h)(0), (h ∈ H∞(Dm))

(Th.2.2.7 de [14]) conformement a proposition 3.2, il existe x′n, y

′n, a

′n et b′n dans H(4) tels que

||x′n||2 + ||y′n||2, ||a′n||2 + ||b′n||2 ≤

√2,

||a′n ⋄T (4)

wj ||, ||wj ⋄T(4)

b′n|| ≤ εn (j = 1, n),

||Ln − (a′n ⋄T (4)

y′n + a′n ⋄T (4)

b′n)|| ≤ εn.

En notant xn :=(T (4)

)nx′n, yn :=

(C(4)

)ny′n, an :=

(C∗

∗(4))n

a′n, bn :=(T ∗(4)

)nb′n, on obtient

que 1√2E0 ∈ Z0. Comme dans la demonstration du theoreme 3.3.11 de [9] (ou voir la demonstration

de theoreme 5.1), on obtient que 1√2Eλ ∈ Z0, pour tout λ ∈ Dm. Donc T (4) a la propriete Z0, 1√

2

Sous la condition supplementaire que T est de type K0,0, on obtient que Z0 ⊂ X0 et donc T (4)

a la propriete X0, 1√2. Conformement au theoreme 2.8 on obtient que T a la propriete X0,1 et donc

Aℵ0aussi, ce qui acheve la demonstration.


5 Multi-contraction a spectredroit de Harte dominant

Le spectre droit de Harte, pour un systeme T = (T1, . . . , Tm) ∈ (L(H))m, commutatif, est

l’ensemble:

σHr (T ) := σH

er ∪ σp(T ∗)

On va voir que l’hypothese de dominance de cet ensemble implique la propriete (A1,ℵ0) pour

le systeme T (2) et sous l’hypothese supplementaire que le systeme T a la propriete K0,., meme lapropriete (Aℵ0

) pour T .

Pour µ = (µ1, . . . , µm) ∈ Dm soit ϕµ la fonction:

ϕµ : Dm → Dm, ϕµ(z1, . . . , zm) =

(z1 − µ1

1− µ1z1, . . . ,

zm − µm

1− µmzm

).

L’application ϕµ est un isomorphisme analitique sur Dm avec ϕ−1µ = ϕ−µ. Nous allons definir

θµ : H∞(Dm) → H∞(Dm), θµ(h) = h ϕµ, (h ∈ H∞(Dm))

L’application θµ est un isomorphisme isometrique f∗−continu. On note πj , (j = 1,m) lesfonctions

πj : Dm → D, πj(z) = zj , (z = (z1, . . . , zm) ∈ Dm)

et

π : Dm → D, π(z) = z1z2 . . . zm, (z = (z1, . . . , zm) ∈ Dm)

Si un m−uplet commutatif de contractions [H, T = (T1, . . . , Tm)] est absolument continu, etΦT est son calcul fonctionnel f∗−continu sur H∞(Dm), alors

ΦT θµ : H∞(Dm) → L(H)

est un morphisme d’algebres de Banach, f∗−continu et avec la propriete ΦT θµ(πj) = ϕµj(Tj).

Donc Tµ := ϕµ(T ) est absolument continu, et ΦTµ= φT θµ. De plus, si T ∈ A alors Tµ ∈ A. La

bijectivite de l’application ϕµ et la relation (Tµ)−µ = T conduisent aux relations:

σHe (Tµ) = ϕµ(σ

He (T ))

etσp(Tµ) = ϕµ(σp(T ))

De plus, comme ϕµ(z) = ϕµ(z) nous aurons que:

ϕµ(σp(T ∗)) = ˜ϕµ(σp(T ∗)) = ˜σp((T ∗)µ) = ˜σp((Tµ)∗)

Si Λ ⊂ Dm est dominant pour Dm alors ϕµ(Λ) a la meme propriete. En effet:

supz∈ϕµ(Λ)

|h(z)| = supz∈Λ

|h ϕµ(z)| = ||h ϕµ||∞ = ||θµ(h)|| = ||h||∞

pour tout h ∈ H∞(Dm).


Pour chaque k ∈ N soit Lk ∈ Q definies comme dans la demonstration de theoreme 4.3. Il estfacile de voir que:

< π2kh, Lk >= h(0)

pour tout h ∈ H∞(Dm).

Dans l’etude de la propriete (A1,ℵ0) pour un systeme T , J. Eschmeier introduit dans [9]

l’ensemble d’approximation Erθ (T ) (ou θ ≥ 0) constitue des elements [f ] de Q pour lesquels il

existe des suites (xn)n≥0 ⊂ (H)1 et (yn)n≥0 ⊂ (K)1 telles que:

limn→∞

||[f ]− xn ⋄C yn|| ≤ θ, (5.1)

limn→∞

||xn ⋄T w|| = 0, (w ∈ H), (5.2)

limn→∞

||w ⋄C yn|| = 0, (w ∈ S). (5.3)

ou [K, C] est une extension co-isometrique minimale de [H, T ] et S est l’espace sur lequel C estcompletement non-unitaire (voir proposition 2.5). On montre que, s’il existe 1 ≥ γ > θ ≥ 0 tel quel’enveloppe absolument convexe fermee de Er

θ (T ) (acof (Erθ (T )) contient la boule fermee de Q de

rayon γ (inclusion qui definit la propriete Erθ,γ pour le systeme T ) alors, T a la propriete (A1,ℵ0

).

Theoreme 5.1. Soit T = (T1, . . . , Tm) ∈ (L(H))m un m−uplet commutatif de contractions, ab-

solument continu tel que (σHe (T ) ∪ σp(T ∗)) ∩Dm soit dominant en Dm. Alors T (2) a la propriete

Er12 ,

1√2

, et en particulier la propriete A1,ℵ0. Si en plus T ∈ K0,. alors T a la propriete X0,1 et donc

(Aℵ0) aussi.

Preuve: Soit (εk)k≥0 ⊂ R∗+ une suite convergente vers zero et wkk≥0 ⊂ H(4) dense. Con-

formement a la proposition 4.2, pour chaque k ∈ N il existe x′k, y

′k ∈ H(4) telles que, pour j = 0, k:

||x′k||2, ||y′k||2 ≤

√2, ||Lk − x′

k⋄T(4)

y′k|| ≤ εk et ||x′k ⋄T (4)

wj || ≤ εk

Si [K, C] est une extension coisometrique minimale de [H, T ] alors [K(4), C(4)] est une extensioncoisometrique minimale de [H(4), T (4)]. Nous allons noter

xk = T (4)kx′k ∈ H(4), yk = C(4)∗ky′k ∈ K(4).

Il existe hk ∈ H∞(Dm) avec ||hk||∞ ≤ 1 tel que:

|| E0 − xk⋄C(4)

yk|| =< hk, E0 − xk⋄C(4)

yk >= hk(0) +

+ < π2khk, Lk − x′k⋄T

(4)y′k > + < π2khk, Lk >≤ ||Lk − x′k⋄T

(4)

y′k|| ≤ εk.

Il resulte limk→∞

||E0 − xk⋄C(4)

yk|| = 0

Si j ∈ N alors pour tout k ≥ j nous avons:

||xk⋄T(4)

wj || =< hk(T(4))T (4)kx′

k, wj >=< πkhk, x′k⋄T

(4)

wj >≤

≤ ||x′k⋄T

(4)

wj || < εk


ou hk ∈ H∞(Dm) avec ||hk||∞ ≤ 1 est donne par le Theoreme de Hahn-Banach. Il resulte que

limk→∞

||xk⋄T(4)

wj || = 0, pour tout j ∈ N. L’ensemble w ∈ H(4) : limk→∞

||xk⋄T(4)

w|| = 0 etant

ferme, il vient:

limk→∞

||xk⋄T(4)

w|| = 0, (w ∈ H(4))

Soit maintenant w ∈ S(4) := w ∈ K(4) : limn→∞

||C(4)nw|| = 0. Il existe hk ∈ H∞(Dm) avec

||hk||∞ ≤ 1 tel que:

||w⋄C(4)

yk|| =< hk(C(4))C(4)kw, y′k >≤ ||C(4)kw||, (k ∈ N)

Cela montre que

limk→∞

||w⋄C(4)

yk||

Les dernieres trois relations montrent que 1√2E0 ∈ Er

0 (T(4)).

Soit maintenant µ = (µ1, . . . , µm) ∈ Dm. Le systeme Tµ est absolument continu et(σHe (Tµ) ∪ ˜σp((Tµ)∗)

)∩

Dm = ϕµ

((σH

e (T ) ∪ σp(T ∗)) ∩Dm)est dominant dans Dm. De plus [K, Cµ] est une extension

coisometrique minimale de [H, Tµ]. Conformement a la premiere etape de la demonstration il

resulte que 1√2E0 ∈ Er

0 ((Tµ)(4)) et donc il existe (ak)k≥0 ⊂

(H(4)

)√√2, (bk)k≥0 ⊂

(K(4)

)√√2telles

que:

limk→∞

||E0 − ak⋄Cµ(4)

bk|| = 0, limk→∞

||ak⋄Tµ(4)

w|| = 0, (w ∈ H(4))

et respectivement, limk→∞

||w⋄Cµ(4)

bk|| = 0, (w ∈ S(4)).

L’application θµ etant un isomorphisme isometrique f∗−continu, il existe γµ : Q → Q isomor-phisme isometrique tel que γµ

∗ = θµ. Pour tout h ∈ H∞(Dm) si x, y ∈ H(4) nous avons:

< h, γ−µ(E0) >= h ϕ−µ(0) = h(µ) =< h, Eµ >

< h, γ−µ(x ⋄Tµ(4)

y) >=< θ−µ(h), x ⋄Tµ(4)

y >=

=< h(T (4))x, y >=< h, x ⋄T (4)

y >

Il resulte que γ−µ(E0) = Eµ si γ−µ(x ⋄Tµ(4)

y) = x ⋄T (4)

y. D’une maniere analogue on obtient

γ−µ(z ⋄Cµ(4)

w) = z ⋄C(4)

w, pour tout z, w ∈ K(4). Il resulte:

limk→∞

||Eµ − ak⋄C(4)

bk|| = limk→∞

||γ−µ

(E0 − ak⋄Cµ

(4)

bk

)|| = 0.

Aussi, pour tout w ∈ H(4) nous avons:

limk→∞

||ak⋄T(4)

w|| = limk→∞

||γ−µ

(ak⋄Tµ

(4)

w)|| = lim

k→∞||ak⋄T−µ

(4)

w|| = 0.

Enfin, pour tout w ∈ S(4) nous obtenons:

limk→∞

||w⋄C(4)

bk)|| = limk→∞

||γ−µ

(w⋄Cµ

(4)

bk))|| = lim

k→∞||w⋄Cµ

(4)

bk)|| = 0.

Ces trois dernieres relations montrent que pour tout µ ∈ Dm:


1√2Eµ ∈ Er

0 (T(4))

Soit maintenant xn = (x1, x2, x3, x4), yn = (y1, y2, y3, y4) ∈ H(4) telles que:

(15) limn→∞

|| 1√2Eλ − xn ⋄T (4)

yn|| = 0

(16) limn→∞

xn ⋄T (4)

w = limn→∞

v ⋄C(4)

yn = 0

pour w ∈ H(4), v ∈ S(4). Comme

4∑

j=1

||xjn|| ||yjn|| ≤

1

2

4∑

j=1

(||xjn||2 + ||yjn||2) ≤ 1

il resulte qu’il existe i, j ∈ 1, 2, 3, 4, i 6= j, et (nk) ⊂ N, tel que

||xink|| ||yink

||+ ||xjnk|| ||yjnk

|| ≤ 1

2.

Pour alleger l’ecriture nous supposons que i = 1 et j = 2. De la relation (15) on obtient:

limk→∞

|| 1√2Eλ −

(x3nk

⋄T y3nk+ x4

nk⋄T y4nk

)|| ≤ 1

2

De la relation (16) il est immediat que

limk→∞

x3nk

⋄T w = limk→∞

x4nk

⋄T w = 0, (w ∈ H)

et

limk→∞

v ⋄C y3nk= lim

k→∞v ⋄C y4nk

= 0, (v ∈ S)

Ces relations montrent que 1√2Eλ ∈ Er

12

(T (2)) pour tout λ ∈ Dm. Donc

(Q) 1√2

= acof 1√2Eλ : λ ∈ Dm ⊂ acof Er

12(T (2))

ou, de maniere equivalente, T (2) a la propriete Er12 ,

1√2

.

Dans l’hypothese que T ∈ K0,., il resulte que S = K et donc Erθ (T ) ⊂ Xθ(T ). Alors T (2) a

la propriete X 12 ,

1√2. Conformement au theoreme 2.8 on obtient que T a la propriete X0,1 et donc

(Aℵ0) aussi.


Bibliographie

[1] E. Albrecht, B. Chevreau, Invariant subspaces for certain representation of H∞(G),Lecture Notes in Pure and Appl. Math., 1994

[2] T. Ando, On a pair of commutative contraction, Acta Sci. MAth., 24(1963), 88-90

[3] C. Apostol, Functional calculus and invariant subspaces, J. Operator Theory 4(1980), 159-190

[4] F.F. Bonsall, J. Duncan, Numerical ranges II, Cambridge University Press, 1973.

[5] S. Brown, Some invariant subspaces for subnormal operators, Integral Equations OperatorTheory,1:310-333, 1978;

[6] S. Brown, B. Chevreau, C. Pearcy, Contraction with rich spectrum have invariant sub-spaces, J. Operator Theory, 1(1979), 123-136.

[7] I. Chalendar, Techniques d’algebres duales et sous-espaces invariants, Monografii Matema-tice, Timisoara 1995.

[8] B. Chevreau, G. Exner, C. Pearcy, On the structure of contraction operators III., Michi-gan Math. J., 36:29-62, 1989.

[9] J. Eschmeier, Invariant subspaces for commuting contractions, J. Operator Theory45(2):413-443, 2001.

[10] D. Gaspar, Analiza Functionala, Ed. Facla, 1980;

[11] B. Sz.- Nagy, C. Foias, Harmonic analysis of operators on Hilbert spaces, North-Holland,Amsterdam, 1970

[12] B. Prunaru, On the structure of contraction operators with dominating spectrum, J. OperatorTheory 20(1988), 42-58.

[13] G. Robel, On the structure of (BCP)-operators and related algebras, J. Operator Theory12(1984), 23-45.

[14] W. Rudin, Function theory in polydisks (Russian translation), 1971

[15] W. Rudin, Functionnal Analysis, Mc. Graw-Hill, 1973.

AURELIAN CRACIUNESCUDepartament de MathematiquesL’Universite de l’Ouest, TimisoaraBv. V. Parvan 41900 Timisoara, RomaniaE-mail:[email protected]




About tensor product stability of some operator ideals

Maria Talpau Dimitriu1

Abstract

We study the tensor product stability of some operator ideals of A− p type.

2010 Mathematics Subject Classification: 47B06, 47L20.Key words: approximation numbers, operator ideals.

1 Indroduction

Let X, Y be two normed spaces and L(X,Y ) the set of all bounded linear operatorsfrom X into Y . The n-th approximation number of T ∈ L(X,Y ) is defined by

an(T ) = inf∥∥T − T

∥∥ : rankT < n, n ∈ N.

For a infinite matrix A = ‖aij‖ and 0 < p <∞, we denote by

LA−p(X,Y ) =

T ∈ L(X,Y ) :

∞∑

i=1

∞∑

j=1

|aij | aj(T )

p

<∞

, 0 < p <∞.

and by LA−p =⋃X,Y

LA−p(X,Y ) the class of operators of A− p type [1], [3].

It can verify that if exists M > 0 such that |ai,2j−1|+ |ai,2j | ≤M |aij | , (∀)i, j ∈ N and

0 <∞∑i=1|ai1|p <∞, then LA−p is an quasinormed operators ideal with quasinorm

‖T‖A−p =

(∞∑i=1

(∞∑j=1|aij |αj(T )

)p) 1p

( ∞∑i=1|ai1|p

) 1p

.

1Department of Mathematics and Informatics, Transilvania University of Brasov, Romania, e-mail:[email protected]

62 Maria Talpau Dimitriu

2 Main result

Lemma 1. Let S, T be two linear and bounded operators, θ a tensorial norm and ϕ :N −→ N a strict increasing function with ϕ(1) = 1. Then

aϕ(n)2 (S ⊗θ T ) ≤ 2 max (‖S‖ , ‖T‖)[aϕ(n) (S) + aϕ(n) (T )

].

In particular, for ϕ(n) = n, n ∈ N we obtain

an2 (S ⊗θ T ) ≤ 2 max (‖S‖ , ‖T‖) [an (S) + an (T )] ,

and for ϕ(n) = 2n−1, n ∈ N we obtain

a22n−2 (S ⊗θ T ) ≤ 2 max (‖S‖ , ‖T‖) [a2n−1 (S) + a2n−1 (T )] .

Proof. It is similar to that for the particular case ϕ(n) = n, n ∈ N, [5].

Theorem 1. If there exists a constant C > 0 such that

maxk∈[j2,j2+2j]

|aik| ≤C

j|aij | , (∀)i, j ∈ N, (1)

then the ideal LA−p is tensor product stable for any tensor norm.

Proof. We have

∞∑

i=1

∞∑

j=1

|aij | aj(S ⊗θ T )

p

1p

=

∞∑

i=1

∞∑

j=1

(j+1)2−1∑

k=j2

|aik| ak(S ⊗θ T )

p

1p

≤

∞∑

i=1

∞∑

j=1

maxk∈[j2,j2+2j]

|aik| aj2(S ⊗θ T )(2j + 1)

p

1p

≤ C

∞∑

i=1

∞∑

j=1

|aij | aj(S) +

∞∑

j=1

|aij | aj(T )

p

1p

≤ C

∞∑

i=1

∞∑

j=1

|aij | aj(S)

p

1p

+

∞∑

i=1

∞∑

j=1

|aij | aj(T )

p

1p

<∞.

About tensor product stability of some operator ideals 63

Example 1. For a matrix A = ‖aij‖, where aij = ai · bj satysfies

∞∑

i=1

|ai|p <∞,

b1 ≥ b2 ≥ · · · > 0, (∃)C > 0, (∀)j ∈ N : bj2 ≤C

jbj ,

we obtain tensor product stable ideals.

Remark 1. For the matrix

A = ‖aij‖ , aij =

αjα1 + · · ·+ αi

, i ≥ j0 , i < j

,

with α1 ≥ α2 ≥ · · · > 0,∞∑n=1

1

(α1 + · · ·+ αn)p< ∞, we obtain the quasinormed ideals of

operators of Stolz − p type [4]

LStolz−p(X,Y ) =

T ∈ L(X,Y ) :

∞∑

i=1

1

α1 + · · ·+ αi

i∑

j=1

αjaj(T )

p

<∞

.

If we suppose that (αn)n satysfies αn2 ≤ C

nαn, (∀)n ∈ N, for some constant C > 0,

then limn→∞

nqαn = 0, (∀)q < 1, [2], and consequently∞∑n=1

1

(α1 + · · ·+ αn)pis a divergent

series since1

(α1 + · · ·+ αn)p≥ 1(

1 +1

2q+ · · ·+ 1

nq

)p and∞∑n=1

1(1 +

1

2q+ · · ·+ 1

nq

)p ∼

∞∑n=1

1

n(1−q)p is divergent for q ≥ 1− 1

p.

If, in definition of the A− p classes, the j-th approximation number is replaced by thej-th dyadic approximation number [5], are obtained more comprehensive classes

L∗A−p(X,Y ) =

T ∈ L(X,Y ) :

∞∑

i=1

∞∑

j=1

|aij | a2j−1(T )

p

<∞

We can prove that

Theorem 2. If A = ‖aij‖ is a infinite matrix such that there exists a constant M > 0 such

that |ai,2j−1|+ |ai,2j | ≤ M |aij | , (∀)i, j ∈ N and 0 <∞∑i=1|ai1|p <∞, then the quasinormed

operators ideal L∗A−p with quasinorm

‖T‖∗A−p =

(∞∑i=1

(∞∑j=1|aij | a2j−1(T )

)p) 1p

( ∞∑i=1|ai1|p

) 1p

64 Maria Talpau Dimitriu

is tensor product stable for any tensor norm.

Proof. We have

∞∑

i=1

∞∑

j=1

|aij | a2j−1(S ⊗θ T )

p

1p

=

∞∑

i=1

∞∑

j=1

[|ai;2j−1| a22j−2(S ⊗θ T ) + |ai;2j | a22j−1(S ⊗θ T )]

p

1p

≤

∞∑

i=1

∞∑

j=1

[|ai;2j−1|+ |ai;2j |] a22j−2(S ⊗θ T )

p

1p

≤ C

∞∑

i=1

∞∑

j=1

|aij | a2j−1(S) +∞∑

j=1

|aij | a2j−1(T )

p

1p

≤ C

∞∑

i=1

∞∑

j=1

|aij | a2j−1(S)

p

1p

+

∞∑

i=1

∞∑

j=1

|aij | a2j−1(T )

p

1p

<∞.

References

[1] B. E. Rhoades, Operators of A − p type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis.Mat. Natur. (8), 59, 3-4(1975), 238-241.

[2] M. Talpau Dimitriu, On the stability of operator ideals with regard to tensor product,Bulletin of the Transilvania University of Brasov, Vol 12(47) (2005), 75-82.

[3] N. Tita, Operators of A− φ type, Atti Accad. Naz. Lincei, 65, 3-4(1978), 97-99.

[4] N. Tita, On Stoltz Mappings, Math. Japonica, 26(1981), 195-196.

[5] N. Tita, On a class of lΦ,ϕ operators, Collect. Math., 32(1981), 275-279.

[6] N. Tita, Lecture notes in “Operator Theory”, Universitatea “Transilvania”, Brasov,1993.

[7] N. Tita, Some interpolation properties and tensor product stability of Stolz map-pings, International Conf. EITM (European Integration Tradition and Modernity,“P. Maior” Univ., Tg. Mures, 2007 (CD), 666-669.




Some special classes of approximation ideals

Adela SASU1

Abstract

Let E, F be a normed spaces and let L(E,F ) be a normed space of all linear andbounded operators T : E → F . We consider the clases:

L∼Φ(E,F ) =

T ∈ L(E,F ) : Φ

(an(T )

1 + an(T )

)< ∞

where Φ is a symmetric norming function and (an(T )) is the sequence of the approx-imation numbers of T .

It is proved that L∼Φ is an ideal in L(E,F ) and some properties of these ideals are

also presented.2000 Mathematics Subject Classification: 47B06, 47L20.Key words: approximation numbers, operator ideals.

1 Introduction

In the paper [3] are introduced the classes of operators Lφ,ϕ(E,F ), defined by thecondition:

Φ (ϕ(an(T )) < ∞where Φ is a symmetric norming function, (an(T )) is the sequence of the approximationnumbers of T ∈ L(E,F ), E and F are normed spaces, and ϕ : [0,∞) → [0,∞), ϕ(0) = 0 isa monotonically increasing function , continuous and subadditive. For ϕ(t) = t one obtainthe classes LΦ, [2].

We recall some of these notions:

(1) K = x = (x1, . . . , xn) ∈ Rn and x1 ≥ x2 ≥ . . . ≥ 0, and the functions Φ : K → Rfulfills:

(i) Φ(x) > 0 if x 6= 0


66 Adela Sasu

(ii) Φ(αx) = αΦ(x), α ≥ 0

(iii) Φ(x+ y) ≤ Φ(x) + Φ(y)

(iv) Φ(1, 0, 0, . . .) = 1

(v) Ifk∑i=1

xi ≤k∑i=1

yi, k = 1, 2, . . . then Φ(x) ≤ Φ(y)

(vi) If x = (x1, x2, . . . , xn, xn+1, . . .) is a sequence, one can consider

Φ(x) = limn→∞

Φ(x1, . . . , xn, 0, 0, . . .)

(2) T ∈ L(E,F ) is defined as the approximation numbers

an(T ) = inf‖T − F‖ : F ∈ L(E,F ), dimF < n

From [5] the following are known:

• ‖T‖ = a1(T ) ≥ a2(T ) ≥ . . . ≥ 0;

• an(λT ) = |λ| · an(T ), λ ∈ R ;

• a2n−1(S + T ) ≤ an(S) + an(T );

• an(ST ) ≤ ‖S‖ · an(T ).

(3) I(E,F ) ⊂ L(E,F ) is an operators ideal if:

(a) S, T ∈ I(E,F ) ⇒ S + T ∈ I(E,F )

(b) S ∈ I(E,F ) and T ∈ L(E,F ) ⇒ TS ∈ I(E,F ) and unidimensional operatorbelongs to I(E,F ).

2 Main Results

We study a special case of the class LΦ,ϕ by considering

ϕ(t) =t

1 + t.

We introduce the class:

L∼Φ(E,F ) =

T ∈ L(E,F ) : Φ

(an(T )

1 + an(T )

)< ∞

(1)

and we show that L∼Φ is an operator ideal in L(E,F ).

We denote the following:

‖T‖∼Φ = Φ

(an(T )

1 + an(T )

). (2)

In the following we check whether the properties of the ideal are fulfilled:

Some special classes of approximation ideals 67

1. If U ∈ L(E,F ) with dimU = 1 then (an(U)) = (‖U‖, 0, 0, . . .) and

‖U‖∼Φ = Φ

( ‖U‖1 + ‖U‖ , 0, 0, . . .

)=

‖U‖1 + ‖U‖ · Φ(1, 0, 0, . . .) =

=‖U‖

1 + ‖U‖ < ∞

Hence, U ∈ L∼Φ .

2. Let S ∈ I∼Φ (E,F ) and T ∈ L(E,F ); one has to prove that TS ∈ I∼Φ (E,F ).

We use: am+n−1(TS) ≤ am(T ) · an(S) ([5]) and by setting m = 1 one obtains:

an(TS) ≤ ‖T‖ · an(S), (∀)n ∈ N. (3)

We have:

‖TS‖∼Φ = Φ

(an(TS)

1 + an(TS)

)≤ Φ

( ‖T‖an(S)1 + ‖T‖an(S)

)≤

≤ Φ

(([‖T‖] + 1)an(S)

1 + ([‖T‖] + 1)an(S)

)≤ ([‖T‖] + 1)‖S‖∼Φ

and hence TS ∈ I∼Φ (E,F ).

3. We prove that if S, T ∈ L∼Φ(E,F ) then S + T ∈ L∼

Φ(E,F )

k∑

n=1

an(S + T )

1 + an(S + T )≤ 2

k∑

n=1

a2n−1(S + T )

1 + a2n−1(S + T )≤ 2

k∑

n=1

an(S) + an(T )

1 + an(S) + an(T )

≤ 2k∑

n=1

(an(S)

1 + an(S)+

an(T )

1 + an(T )

)

and thus

Φ

(an(S + T )

1 + an(S + T )

)≤ 2

[Φ

(an(S)

1 + an(S)

)+Φ

(an(T )

1 + an(T )

)]

⇒ ‖S + T‖∼Φ ≤ 2 (‖S‖∼Φ + ‖T‖∼Φ)If we consider the following functions:

Φ(p)(x) = (Φ(xpi ))1p , 1 ≤ p ≤ ∞

then they are simmetric norming functions (see [5]) if Φ is also such a function.Therefore, one can consider the classes:

L∼Φ,p(E,F ) =

T : ‖T‖∼Φ,p = Φ(p)

(an(T )

1 + an(T )

)< ∞

(4)

By taking into account that Φ(p2)(x) ≤ Φ(p1)(x), (∀)x ∈ K, 1 ≤ p1 ≤ p2 < ∞ we getthat one can obtain inclusion relationships among these classes.

68 Adela Sasu

Proposition 1. L∼Φ,p1

⊂ L∼Φ,p2

, 0 ≤ p1 < p2 < ∞. (see [5])

Remark 1. By varying the form of function Φ, one obtains various ideals from LΦ,ϕ,e.g.:

L∗Φ,ϕ(E,F ) =

T :

(∑αn(ϕ(an(T ))p

) 1p< ∞

(5)

For the class defined by (5) we observe that in the paper “Representation Theorems

for Operators of type lv,w,ψp,q and Sω,ψ” published in GLASNIK MATEMATICKI. Vol.43(63)(2008), 423–437, similar classes appears, with a particular form of αn, and hencethey are particular cases of L∗

Φ,ϕ.

In [4] it is shown that

1

1 + ‖T‖an(T ) ≤an(T )

1 + an(T )≤ an(T ), (∀)n ∈ N. (6)

Thus1

1 + ‖T‖Φ(an(T )) ≤ Φ

(an(T )

1 + an(T )

)≤ Φ(an(T )) (7)

Therefore

‖T‖Φ · 1

1 + ‖T‖ ≤ ‖T‖∼Φ ≤ ‖T‖Φ (8)

that is

‖T‖Φ∼‖T‖∼Φ (9)

from which one can obtain other properties of classes L∼Φ .

From [2] and [5] we recall that

‖ST‖Φ ≤ 2‖S‖Φ(p) · ‖T‖Φ(q), (10)

where 1p +

1q = 1 and we obtain

‖ST‖∼Φ ≤ ‖ST‖Φ ≤ 2‖S‖Φ(p) · ‖T‖Φ(q) ≤ 2(1 + ‖S‖) · (1 + ‖T‖)‖S‖∼Φ(p) · ‖T‖∼Φ(q) (11)

where 1p +

1q = 1.

From eq. 6 we obtain:

Proposition 2. If S = L∼Φ(p) and T ∈ L∼

Φ(q) then ST ∈ L∼Φ.

References

[1] Gohberg, I., Krein, M., Introducere ın teoria operatorilor liniari neautoadjuncti ınspatii Hilbert (lb. rusa), Nauka-Moscova, 1965.

[2] Tita, N., Operatori de clasa σp pe spatii nehilbertiene, Studii. Cercet. Mat. 23, p.467-487, 1971.

Some special classes of approximation ideals 69

[3] Tita, N., Lφ,ϕ operators and (φ, ϕ) spaces, Seminario Mat. De Barcelona, (CollectMath) 30, p. 3-10, 1979.

[4] Tita, N., Note de curs la “Teoria operatorilor”, Universitatea Transilvania Brasov,1993.

[5] Tita, N., Ideale de operatori generate de s-numere, Editura Universitatii TransilvaniaBrasov, 1998.

[6] Schatten, R., Norm ideals of completely continuous operators, Springer Verlag Berlin,1960.



Remarks on some classes of linear operators

Nicolae Tita1

Abstract

We remark that the subject analysed in the paper [1] has been studied also in thepapers [2], [3] etc. Other remarks are also presented.

AMS Subject Classification: 47B06, 47L20.Key Words and Phrases: linear operators, approximation numbers, s-numbers,

operator ideals.

1 Indroduction

Let X, Y be two normed spaces and T : X −→ Y is a linear and bounded operator(T ∈ L(X,Y )). The sequence of the approximation numbers of T is (an(T )), an(T ) =inf∥∥T − T

∥∥ : rankT < n, n = 1, 2, . . .. In [1] are considered also other s-numbers

sequences. All these sequences have the so called additivity properties:

sm+n−1(S + T ) ≤ sm(S) + sn(T ), m, n = 1, 2, . . . .

The purpose of the paper [1] is to analyse the classes of “A− p type operators“.

Recall that T is an operator of A − p class if, for a matrix A = ‖aij‖, the relation∞∑i=1

(∞∑j=1

|aij | aj(T ))p

< ∞ is true, 0 < p < ∞. If aij =1

ifor i ≥ j and aij = 0 for j > i,

it results the Cesaro matrix and respectively the class of Ces− p operators; T ∈ Ces− p

if∞∑i=1

(1

i

i∑j=1

aj(T )

)p

< ∞.

In 1974, K. Iseke has replaced the arithmetic mean1

i

i∑j=1

aj(T ) by the mean

1

α1 + · · ·+ αi

i∑j=1

αjaj(T ), where α1 ≥ α2 ≥ · · · ≥ 0 and∞∑i=1

αi = ∞. In this way the class

of Stolz operators is a special case of A− p type operators.


Remarks on some classes of linear operators 71

If aij = δij =

1 , if i = j

0 , if i 6= j, results the well known classes lp, defined by the condition:

∞∑n=1

apn(T ) < ∞, 0 < p < ∞.

2 Remarks

In the paper [3] is proved the following:

Theorem 1. [3] The class of Stolz mapping coincides with the class lp, 1 < p < ∞,if lim

n→∞αn 6= 0.

Corollary 2. [3] The class Ces− p coincides with the class lp, 1 < p < ∞.

In conclusion, the ideals Ces − p coincide with the ideals lp, 1 < p < ∞ and theseideals are well known.

Relative to a negative answer to a problem related to A − p operators it can see also[2].

Also, the inequality:k∑

i=1si(S+T ) ≤ 2

k∑i=1

(si(S) + si(T )) , k = 1, 2, . . . , from [1] p. 736

is my inequality: Studii Cercet. mat. 23(1971) 467-487. It can see [4], [5], [6] etc.

3 Other remarks and results

Since Φα : (x1, . . . , xn) −→n∑

i=1αixi is a symmetric norming function, if x1 ≥ x2 ≥ · ≥ 0,

1 = α1 ≥ α2 ≥ · ≥ 0, in [7] the arithmetic mean

n∑i=1

αixi

n∑i=1

αi

has been replaced by a Φ-

arithmetic mean MΦ, where MΦ (xi) =Φ (xi)Φ(n)

; Φ is a symmetric norming function and

Φ(n) = φ(1, 1, . . . , 1︸︷︷︸n−times

, 0, 0, . . .).

In this way the class LMΦ,p =

T :

∞∑n=1

[Φ (si(T )n1 )

Φ(n)

]p< ∞

is a generalization of

the class of Stolz mappings, which is of interest when limn→∞

αn = 0 or when limn→∞

αn 6= 0

but 0 < p ≤ 1.We can prove that LMΦ,p ⊂ Ces − p, for all Φ and 0 < p < ∞. This is a corollary of

the following proposition.Firstly we recall the definition of a symmetric norming function (s.n.f.), [2], [4], [5], [6],

[7].Let be K = x = (x1, . . . , xn) ∈ Rn : x1 ≥ x2 ≥ · ≥ 0. Φ : K −→ R+ and satisfies

the following properties: (1)Φ(x) > 0 if x 6= 0; (2)Φ(αx) = αΦ(x), ∀α ≥ 0; (3)Φ(x+y) ≤

72 Nicolae Tita

Φ(x) + Φ(y); (4)Φ(1, 0, 0, . . .) = 1; (5) If x, y ∈ K andn∑

i=1xi ≤

n∑i=1

yi, k = 1, 2, . . . , n, then

Φ(x) ≤ Φ(y).

Now it can prove:

Proposition 3. The inequality

n∑i=1

xi

n≤ Φ (xin1 )

Φ(n)is true, for all s.n.f. and x =

(xi) ∈ K.

Proof. The inequality is Φ(n)n∑

i=1xi ≤ Φ (nxin1 ), or

Φ(n∑

i=1

xi, . . . ,n∑

i=1

xi

︸︷︷︸n−times

, 0, 0, . . .) ≤ Φ (nxin1 ). This is true if: kn∑

i=1xi ≤ n

k∑i=1

xi, k = 1, 2, . . . , n.

But kn∑

i=1xi = k

k∑i=1

xi+kn∑

i=k+1

xi and nk∑

i=1xi = k

k∑i=1

xi+(n−k)k∑

i=1xi. This means that

the inequality is true if kn∑

i=k+1

xi ≤ (n− k)k∑

i=1xi. Since (xi) ∈ K results that k

n∑i=k+1

xi ≤

(n− k) · k · xk+1 ≤ (n− k)k∑

i=1xi. This prove the inequality, because k · xk+1 ≤

k∑i=1

xi.

4 Final remarks

From the proposition results the desired inclusion (in particular Stolz mappings ⊆ Ces−p),but are some other applications.

If y = (yi) ∈ K is such that y1 = 1, then Φy(x) =n∑

i=1xiyi is a s.n.f. and from

proposition results the Chebyshev inequality:1

n

(n∑

i=1xi

)·(

n∑i=1

yi

)≤

n∑i=1

xiyi.

Also, if Φ∗ is the dual function of Φ

Φ∗(y) = sup

x∈K,x 6=0

n∑i=1

xiyi

Φ(x)

we obtain n = Φ(n) ·

Φ∗(n). For the particular case when Φ is Φp : (xi) −→n∑

i=1(xpi )

1p , 1 ≤ p < ∞, it is known

that (Φp)∗ ≡ Φq, 1 =

1

p+

1

q, and n = Φp(n)Φq(n).

References

[1] P. D. Srivastava, A. Maji, On some class of operator ideals, Internat. J. Pure andAppl. Math., 83, No. 5(2013), 731-740.

Remarks on some classes of linear operators 73

[2] N. Tita, Operators of A− φ type, Atti Accad. Naz. Lincei, 65, 3-4(1978), 97-99.

[3] N. Tita, On Stoltz Mappings, Math. Japonica, 26(1981), 195-196.

[4] N. Tita, Ideale de operatori generate de s-numere, Ed. Univ. “Transilvania” Brasov,1998.

[5] N. Tita, Equivalent quasinorms on some Approximation Ideals, Bull. Acad. st Rep.Moldova (Izvestia AN Rep Mold.), matematica, 2(33) (2000), 60-66.

[6] N. Tita, A general view on Approximation Ideals, Functional Analysis and its Appl.North - Holland Math. Stud. 197(2004), 295-300.

[7] N. Tita, Some interpolation properties and tensor product stability of Stolz map-pings, International Conf. EITM (European Integration Tradition and Modernity,“P. Maior” Univ., Tg. Mures, 2007 (CD), 666-669.




Some remarks on the infinite-variate prediction

Ilie Valusescu 1

AMS Classification: 47N30, 60G25, 47A20, 62H99

Keywords: L2-bounded analytic functions, factorization theorems, complete corre-lated actions, operator model, Γ-correlated processes, Γ-orthogonal projection, infinite-variate prediction.

1 Introduction

In the study of stationary processes a central role is played by the factorization theorems.For the univariate case Kolmogorov [6] gave his elegant solution for the prediction problemarised by Khintchine [5], using the result of Szego [15] about representation f(t) = |q(eit|2of a scalar valued function f(t) ≥ 0 by means of a scalar valued analytic function q(λ)on the unit disc D. In the multivariate (finite) case, the same role was played by the fac-torization theorems for matrix valued functions, begining with Zasuhin [19] and stronglycontinuated by Wiener [20], Wiener and Masani [21], Helson and Lowdenslager [4]. Thesame central role in the study of prediction problems was played by the Wold decomposi-tion theorems in various generalized forms, starting with Wold [22] for univariate case.

In a natural way, for the prediction theory of a stationary process with infinitely manycomponents, factorization theorems and Wold-type decompositions for operator valuedfunctions are necessary. Such theorems arised, but in their generalities, new difficultiesrelated to the boundary values of an operator valued function in the unit disc appeared.To be mentioned here the results of Devinatz [1] and Lowdenslager [7].

In the case of bounded analytic functions, B.Sz.-Nagy and C. Foias [16] avoided thesedifficulties by using the bounded convergence principle in order to construct the boundaryfunction in the Fatou theorem about non-tangentially a.e. convergence (in strong sense)and consequently, they obtained totally satisfactory factorization theorems in the boundedcase, but these are not sufficient to solve the infinite-variate prediction problem.

Some prediction theoretically properties of B.Sz.-Nagy and C. Foias model and con-nections between them and factorization theorems for operator valued functions given by

1The work was supported by UEFISCDI Grant PN-II-ID-PCE-2011-3-0119 Institute of Mathemat-ics ”Simion Stoilow” of the Romanian Academy, Calea Grivitei nr. 21, Bucharest, Romania; e-mail:[email protected]

Some remarks on the infinite-variate prediction 75

Devinatz [1] and Lowdenslager [7] were established, but, partly due to the fact that inthe non-bunded case the proof of Lowdenslager’s theorem was not totally satisfactory andpartly since in a time-domain analysis of infinite variate stationary processes and espe-cially prediction problems were not clearly formulated, these theoretical properties werenot sufficiently used in order to give a consistent prediction theory in infinite-variate case.

Since Douglas [2] given an example which shows that the result of Lowdenslager isnot valid in the bounded case, the research to extend to the infinite-variate predictionwas locked for a while. But Douglas’ statements are somewhat unnatural from predictiontheory point of view, he involve boundedness assumptions which, a priori, need not besatisfied. To circumvent these difficulties, in [11] the Lowdenslager theorem is made clearin its full generality in order to be used in prediction theory. The main idea in [11]was to enlarge the class of operator valued functions from the bounded case to the L2-bounded one, and to construct instead of a boundary function, a semispectral measurewhose Poisson integral is the considered L2-bounded function. So, the Lowdenslagertheorem is still valid from prediction point of view, and a consistent model for infinitevariate prediction was obtained, as it is mentioned in [8].

This paper is mainly an expository one, presenting a way to obtain a consistent for-mulation for prediction problems in the infinite-variate case, and how a linear filter forprediction can be found in the stationary discrete case. The main results, as the gener-alized factorization theorem, Wold decomposition, the linear filter for discrete stationaryprocesses, are obtained in [11, 12, 14], and for the nonstationary case are obtained by theauthor [17, 18], especially for the periodically correlated case.

2 Factorization theorem

Let E be a separable Hilbert space, D and T be the unit disc and the unit torus in thecomplex plane C. As usually, the set of real, integer and natural numbers are, respectively,denoted by R,Z,N, and the C∗-algebra of all linear bounded operators on a Hilbert spaceE will be denoted by L(E). For the beginning, let us recall some necessary known facts.

An L(E)-valued semispectral measure on T is a map F : B(T) → L(E) such thatσ → (F (σ)a, a) is a positive Borel measure.

A semispectral measure E is spectral if E(σ1∩σ2) = E(σ1)·E(σ2) for any σ1, σ2 ∈ B(T),and E(T) = IE.

By Naimark dilation theorem, for any L(E)-valued semispectral measure F there existsa spectral dilation [K, V, E], i.e. K is a Hilbert space, V ∈ L(H,K), and E is an L(K)-valued spectral measure on T such that F (σ) = V ∗E(σ)V .

The spectral dilation is minimal if

K =∨

σ∈B(T)

E(σ)V E.

For a Hilbert space F, we denote by E×F the spectral measure corresponding to the

multiplication by eit on L2(F). An L(E)-valued semispectral measure F on T is of analytictype if it admits a spectral dilation of the form [L2(F), V, E×

F ] such that V E ⊂ H2(F). The

76 Ilie Valusescu

name is justified because there exists an operator valued analytic function E,F,Θ(λ) onD, i.e.

Θ(λ) =∞∑

n=0

λnΘn, Θn ∈ L(E,F),

such thatΘ(λ)a = (V a)(λ); (λ ∈ D).

For our purpose we need to enlarge the class of bounded analytic functions E,F,Θ(λ)on D to unbounded one, as follows:

An operator valued analytic function E,F,Θ(λ) on D is an L2-bounded analyticfunction if there exists a constant M > 0 such that for any a ∈ E

∞∑

k=0

‖Θka‖2 ≤ M‖a‖2, (2.1)

or equivalently,

sup0≤r<1

1

2π

2π∫

0

‖Θ(reit)a‖2dt ≤ M‖a‖2. (2.2)

The operator valued analytic function E,F,Θ(λ) is bounded if

‖Θ(λ)‖ ≤ M, (λ ∈ D). (2.3)

Remark: Obviously a bounded function is L2-bounded, but the converse is no longervalid. The unpleasant fact is that L2-bounded functions have no boundary limits Θ(eit)and a lot of very nice properties of contractive functions can not be proved using this fact.

To surpass this fact, to any L2-bounded E,F,Θ(λ) we attach VΘ ∈ L(E, H2(F)) by

(VΘa)(λ) = Θ(λ)a, (a ∈ E), (2.4)

and conversely, for V ∈ L(E, H2(F)) we obtain an L2-bounded analytic function E,F,Θ(λ)by

Θ(λ)a = (V a)(λ), λ ∈ D.

PuttingFΘ(σ) = V ∗

ΘE×F (σ)VΘ, σ ∈ B(T), (2.5)

we can attach to each L2-bounded analytic function E,F,Θ(λ) a semispectral measureof analytic type FΘ.

Similar like in the bounded case, the inner and outer functions in the L2-bounded caseare defined.

An L2-bounded analytic function E,F,Θ(λ) is an inner function, if it is boundedand Θ(eit) is an isometry a.e. on T.

An L2-bounded analytic function E,F,Θ(λ) is an outer function, if

∞∨

0

eintVΘE = L2+(F). (2.6)


Obviously, like in [16], it can be proved that each outer L2-bounded analytic function Θ(λ)has the property

Θ(λ)E = F, (λ ∈ D), (2.7)

and if an L2-bounded analytic function is simultaneously inner and outer, then it is aunitary constant function.

To prove the generalized Lowdenslager–Sz-Nagy–Foias theorem, which will play a cru-cial role in generalization of the prediction problems to the infinite-dimensional case, firstlyit is proved (see [11]) the following factorization theorem, in a similar manner to [16].

Theorem 2.1. Let E,F,Θ(λ) and E,F1,Θ1(λ) be two L2-bounded analytic functions,the second one being outer, FΘ and FΘ1 be the corresponding semispectral measures. Sup-pose that

FΘ ≤ FΘ1 . (2.8)

Then there exists a contractive analytic function F1,F,Θ2(λ) such that

Θ(λ) = Θ2(λ)Θ1(λ) (λ ∈ D). (2.9)

If in (2.8) the equality holds, then F1,F,Θ2(λ) is inner. If moreover E,F,Θ(λ) isouter, then F1,F,Θ2(λ) is unitary constant.

Theorem 2.2. (Lowdenslager, Sz.-Nagy, Foias factorization theorem). Let F be an L(E)-valued semispectral measure on T and [K, V, E] be its minimal spectral dilation. Thereexists a unique L2-bounded outer function E,F1,Θ1(λ) with the properties:

(1) FΘ1 ≤ F

(2) For any L2-bounded analytic function E,F,Θ(λ) such that FΘ ≤ F we have also

FΘ ≤ FΘ1 .

The properties (1) and (2) determines the outer function Θ1 up to a left unitary con-stant factor.

The equality holds in (1) if and only if

⋂

n≥0

UnK+ = 0, (2.10)

where U is the unitary operator on K corresponding to the spectral measure E and

K+ =∞∨

n=0

UnV E.

Proof. Let U+ be the restriction of the unitary operator U on K+. The operator U+ beingan isometry on K+ we can consider its Wold decomposition

K+ =( ∞⊕

0

UnF1

)⊕ R,

78 Ilie Valusescu

where F1 = K+ ⊖ U+K+ and R =⋂n≥0

Un+K+.

Because R is a reducing space for U we have also

K =( +∞⊕

−∞UnF1

)⊕ R. (2.11)

The space+∞⊕−∞

UnF1 is a reducing subspace for U , and, if P is the orthogonal projection

of K onto+∞⊕−∞

UnF1, then PU = UP .

Let ΦF1 be the canonical isomorphism between+∞⊕−∞

UnF1 and L2(F1), and VΘ1 be the

linear bounded operator from E into L2(F1) given by

VΘ1a = ΦF1PV a (a ∈ E). (2.12)

Then VΘ1E ⊂ L2+(F1) and

∞∨

0

eintVΘ1E =∞∨

0

eintΦF1PV E = ΦF1

∞∨

0

UnPV E =

= ΦF1P

∞∨

0

UnV E = ΦF1PK+ = ΦF1( ∞⊕

0

UnF1

)= L2

+(F1).

Therefore the L2-bounded analytic function E,F1,Θ1(λ) corresponding to VΘ1 is outer.For any analytic polynomial p we have

∫ 2π

0

∣∣p(eit)∣∣2 d(FΘ1(t)a, a)E = ‖pVΘ1a‖2L2(F1)

=∥∥∥pΦF1PV a

∥∥∥2

L2(F1)=

= ‖p(U)PV a‖2K = ‖Pp(U)V a‖2K ≤ ‖p(U)V a‖2K =

∫ 2π

0

∣∣p(eit)∣∣2 d(F (t)a, a)E.

Thus FΘ1 ≤ F . The equality holds if and only if ΦF1PV = ΦF1V , i.e. if and only ifPV = V , i.e. if and only if R = 0, i.e. if and only if

⋂n≥0

UnK+ = 0.

If E,F,Θ(λ) is another L2-bounded analytic function such that FΘ ≤ F , then for

the elements of K of the formn∑

k=0

UkV ak we consider

X( n∑

k=0

UkV ak)=

n∑

k=0

eiktVΘak. (2.13)

Using again the complete positivity of the semispectral measures we have∥∥∥∥∥X

( n∑

k=0

UkV ak)∥∥∥∥∥

2

=n∑

k,j=0

∫ 2π

0ei(k−j)td(FΘ(t)ak, aj) ≤


≤n∑

k,j=0

∫ 2π

0ei(k−j)td

(F (t)ak, aj

)=

∥∥∥∥∥n∑

k=0

UkV ak

∥∥∥∥∥

2

.

It follows that (2.13) gives rise to a contraction X from K+ into L2+(F) such that XU =

eitX and we have

XR = X⋂

n≥0

UnK+ ⊂⋂

n≥0

XUnK+ =⋂

n≥0

eintXK+ ⊂⋂

n≥0

eintL2+(F) = 0.

Hence XP = X, and for any analytic polynomial p we have

∫ 2π

0

∣∣p(eit)∣∣2 d(FΘ(t)a, a) = ‖pVΘa‖2 = ‖Xp(U)V a‖2 = ‖XPp(U)V a‖2 ≤ ‖Pp(U)V a‖2 =

=∥∥∥ΦF1Pp(U)V a

∥∥∥2=

∥∥∥p(eit)ΦF1PV a∥∥∥2= ‖pVΘ1a‖2 =

∫ 2π

0

∣∣p(eit)∣∣2 d(FΘ1(t)a, a)

and it follows that FΘ ≤ FΘ1 .Now, for any L2-bounded outer function E,F′

1,Θ′1(λ) which satisfies the conditions

(1) and (2) we have FΘ′1= FΘ1 , and using Theorem 2.1 it results that Θ′

1(λ) = ZΘ2(λ),where Z is a unitary operator from F′

1 to F1.The proof is finished.

The previous theorem generalize the Sz.-Nagy–Foias factorization theorem [16]. Wehave only to take dF (t) = N(t)2d(t) for 0 ≤ t ≤ 2π, where N(t) is a function whosevalues are self-adjoint operators on a separable Hilbert space E, and which is measurable(strongly or weakly, whivh amounts to the same for separable E) with the property that0 ≤ N(t) ≤ I, then the requested result is obtained. Indeed, for any trigonometricpolynomial p we have

∫ 2π

0

∣∣p(eit)∣∣2 ∥∥(VΘ1a)(e

it)∥∥2 dt = ‖pVΘ1a‖2 =

∫ 2π

0

∣∣p(eit)∣∣2 d(FΘ1(t)a, a) ≤

≤∫ 2π

0

∣∣p(eit)∣∣2 d(F (t)a, a) =

∫ 2π

0

∣∣p(eit)∣∣2 ‖N(t)a‖2 dt.

Thus we deduce that∥∥(VΘ1a)(e

it)∥∥2 ≤ ‖N(t)a‖2 ≤ ‖a‖2 a.e., and it follows that Θ1(λ)

is a contractive outer function.Suppose now that Θ1(λ) has non-tangentially boundary limits Θ1(e

it) a.e. (as stronglimits of operators) and, moreover, that in (1) we have equality. Then we have

dF (t) = Θ1(eit)∗Θ1(e

it)dt, (2.14)

which corresponds to the usual meaning of factorability of measures by means of analyticfunctions.

Due to the maximality property (2) from Theorem 2.2 of the semispectral measure FΘ1

between all analytic type semispectral measures dominated by F , the unique L2-bounded

80 Ilie Valusescu

outer function E,F1,Θ1(λ) is called the maximal outer function, or shorter, the maximalfunction of F . Such a way, to each semispectral measure F a maximal function Θ1(λ) isattached.

Using Theorem 2.1 and Theorem 2.2, the canonical Beurling factorization, i.e. thefactorization into a product of an inner function and an outer one, can be derived forL2-bounded analytic functions as follows.

Theorem 2.3. Every L2-bounded analytic function E,F,Θ(λ) can be uniquely factorizedinto the form

Θ(λ) = Θi(λ)Θe(λ) (λ ∈ D), (2.15)

where E,F1,Θe(λ) is an outer L2-bounded analytic function and F1,F,Θi(λ) is aninner function.

Szego theorem [15] and its implications in factorizations plays an important role inprediction theory as in various area of mathematics. In operator theory this theorem isalso very intimately related with basic problems like the structure of invariant subspaces,Jordan models, cyclicity, etc. The applications of the operatorial methods in predictiontheory cross also through ideas contained in this very important theorem.

Following the treatement given in [16] for the bounded operator valued analytic func-tions, a variant of the Szego theorem which is usefull in operatorial prediction was obtained[13] in the L2-bounded case.

To do this, a Szego operator was attached to each semispectral measure F as follows.

Let F be an L(E)-valued semispectral measure and [K, V, E] be its minimal spectraldilation. If we put

(∆[F ]a, a

)= inf

n∑

k,j=0

∫ 2π

0ei(k−j)td(F (t)ak, aj), (2.16)

where the infimum is taken over all finite systems of elements a0, a1, . . . , an from E suchthat a0 = a, then

(∆[F ]a, a

)= inf

n∑

k,j=0

∫ 2π

0ei(k−j)td(F (t)ak, aj) = inf

n∑

k,j=0

∫ 2π

0ei(k−j)td(E(t)V ak, V aj) =

= infn∑

k,j=0

(Uk−jV ak, V aj) = inf

∥∥∥∥∥∥V a−

n∑

k,j=0

UkV ak

∥∥∥∥∥∥

2

= ‖(I − P1)V a‖2 =(V ∗(I−P1)V a, a

),

where U is the unitary operator corresponding to the spectral measure E and P1 is the

orthogonal projection of K onto the subspace∞∨1UkV E. Hence

∆[F ] = V ∗(I − P1)V. (2.17)


Therefore ∆[F ] is a positive operator on E which will be called the Szego operator of theL(E)-valued semispectral measure F . The name is justified by the fact that in the scalarvalued case F = µ we have

∆[µ] = infp0

∫ 2π

0|1− p0|2 dµ (2.18)

where the infimum is taken over all analytic polynomials p0 which are vanishing in theorigin.

So, the following generalization of the Szego-Kolmogorov-Krein theorem is obtained.

Theorem 2.4. Let F be an L(E)-valued semispectral measure on T and ∆[F ] be the cor-responding Szego operator. Then

(i) ∆[F ] = 0 if and only if there exist no non-null L2-bounded analytic functionE,F,Θ(λ), such that FΘ ≤ F .

(i) If ∆[F ] 6= 0, then there exists a unique maximal outer L2-bounded analytic functionE,F1,Θ1(λ) such that FΘ1 ≤ F, dimF1 = dim(∆[F ]E) and

∆[F ] = ∆[FΘ1 ] = Θ1(0)∗Θ1(0). (2.19)

Proof. If E,F,Θ(λ) is an L2-bounded analytic function and E,F1,Θ1(λ) its outer part,then FΘ1 ≤ FΘ and

(∆[F ]a, a

)=

(∆[FΘ1 ]a, a

)= inf

a1,...,an∈E

∥∥∥∥∥VΘ1a−n∑

k=1

eiktVΘ1ak

∥∥∥∥∥

2

=

= inff∈F1

‖VΘ1a− f‖2 =∥∥V(Θ1

a)(0)∥∥2 = ‖Θ1(0)a‖2 .

Hence, if FΘ1 ≤ F , then Θ1(0)∗Θ1(0) = ∆[FΘ1 ] ≤ ∆[F ].

Now, let E,F1,Θ1(λ) be the maximal outer function of the semispectral measure

F . If P is the orthogonal projection of K on∞⊕−∞

UnF1, then (I − P1)P = P (I − P1) =

I−P1. Therefore (∆[F ]a, a) = ‖(I − P1)V a‖2 = ‖(I − P1)PV a‖2 = ‖(P − P1P )PV a‖2 =(∆[FΘ1 ]a, a). It follows that ∆[F ] = ∆[FΘ1 ] = Θ1(0)

∗Θ1(0) and taking account that forany λ ∈ D we have Θ(λ)E = F, and the proof is finished.

3 Prediction problems

To extend the study of prediction theory from the finite multivariate case to the infinitevariate one, it was necessary to come out from the very convenient frame of a Hilbertspace and to work into a more general context of a module, as follows.

Let E be a Hilbert space and H be a right L(E)-module. By an action of L(E) on H

we mean the map from L(E) × H into H given by Ah := hA in the sense of the rightL(E)-module H. We are writting Ah instead of hA to respect the classical notations from

82 Ilie Valusescu

the scalar case. A correlation of the action of L(E) on H is a map Γ from H × H intoL(E) having the properties:

(i) Γ[h, h] ≥ 0, and Γ[h, h] = 0 implies h = 0;(ii) Γ[h, g]∗ = Γ[g, h];(iii) Γ[h,Ag] = Γ[h, g]A.A triplet E,H,Γ defined as above was called a correlated action of L(E) on H.By the fact that generally in H we have no topology, the prediction subsets, such as

past and present, future, etc., can not be seen as closed subspaces, therefore the powerfultool of the orthogonal projection can not be directly used.

An example of correlated action can be constructed as follows. Take as the right L(E)-module H = L(E,K) – the space of the linear bounded operators from E into K, where E

and K are Hilbert spaces. An action of L(E) on L(E,K) is given if we consider AV := V Afor each A ∈ L(E) and V ∈ L(E,K). It is easy to see that Γ[V1, V2] = V ∗

1 V2 is a correlationof the action of L(E) on L(E,K), and the triplet E,L(E,K),Γ is a correlated action (theoperator model). It was proved [12] that any abstract correlated action E,H,Γ can beembedded into the operator model. Namely, there exists an algebraic embedding h → Vh

of H into L(E,K), where K is obtained as the Aronsjain reproducing kernel Hilbert spacegiven by a positive definite kernel obtained from the correlation Γ. The generators of Kare elements of the form γ(a,h) : E × H → C, where γ(a,h)(b, g) = 〈Γ[g, h]a, b〉E and theembedding h → Vh is given by Vha = γ(a,b).

Due to such an embedding of any correlated action E,H,Γ into the operator model,prediction problems can be formulated and solved using operator techniques. In the par-ticular case when the embedding h → Vh is onto, the correlated action E,H,Γ is caleda complete correlated action.

In the following the Hilbert spaceK uniquely attached to the correlated action E,H,Γwill be called the measuring space of the correlated action. The name is justified by thefact that having a state h in the state space H, what we can measure is the elementVha from the Hilbert space K. In prediction problems we are inerested in measuring thecloseness between two states, and this fact is not possible to be directely made in the statespace H which is only a right L(E)-module, but it is possible to be done in the measuringspace K, and must be interpreted in H. So, we need to have the possibility to ”interpret”each element from K in terms of the state space H. This fact implies a completenesscondition imposed to the algebraic imbedding of H into L(E,K). In [10], some ideas foran axiom of completeness are given, such that the prediction problems to be solved. Inthis paper most of properties are analysed in the complete correlated case.

For prediction purposes, especially to obtain the best estimation, we need to have akind of ”projection” on a subset of the state space H. Because H is only a right L(E)-module, we can not have a projection on a submodule. But in the case of a completecorrelated action of L(E) on H, a ”Γ-orthogonal projection” on a submodule H1 of H canbe defined, as it follows from the following Proposition.

Proposition 3.1. Let H1 be a submodule in the right L(E)-module H and

K1 =∨

x∈H1

VxE.


For each h ∈ H there exists a unique element h1 ∈ H such that for each a ∈ E we have

Vh1a ∈ K1 and Vh−h1a ∈ K⊥1 . (3.1)

Moreover, we have

Γ[h− h1, h− h1] = infx∈H1

Γ[h− x, h− x], (3.2)

where the infimum is taken in the set of all positive operators from L(E).

Proof. If PK1 is the orthogonal projection of the Hilbert space K on the subspace K1 and

Vh1 = PK1Vh, (3.3)

then for any a ∈ E the element Vh1a is in K1. Also

Vh−h1a = Vha− Vh1a = Vha− PK1Vha = (I − PK1)Vha ∈ K⊥1 .

If h2 is an element from H with the property (3.1), then for any a ∈ E we have

Vha = Vh2a+ Vh−h2a.

It follows that Vh2a = PK1Vha, hence h2 = h1.

We also have

(Γ[h− h1, h− h1]a, a

)= ‖Vh−h1a‖2 = ‖(I − PK1)Vha‖2 = inf

k∈K1

‖Vha− k‖2 =

= infn∑1

∥∥∥∥∥Vha−n∑

1

Vxjaj

∥∥∥∥∥

2

= infn∑1

∥∥∥∥∥∥Vha− V n∑

1Ajxj

a

∥∥∥∥∥∥

2

=

= infn∑1

(Γ[h−

n∑

1

Ajxj , h−n∑

1

Ajxj ]a, a)= inf

x∈H1

(Γ[h− x, h− x]a, a

),

where for any finite systems (a1, . . . , an) of elements from E we choose a system of operators(A1, . . . , An) in L(E) such that Aja = aj , j = 1, . . . , n, and the proof is finished.

If we put

PH1h = h1,

then clearly we obtain an endomorphism of H such that

VP2H1

h = VPH1h1 = PK1Vh1 = P 2

K1Vh = PK1Vh = VPH1

h

and also

Γ[PH1h, g] = V ∗PH1

hVg = (PK1Vh)∗Vg = V ∗

h PK1Vg = V ∗h VPH1

g = Γ[h,PH1g].

84 Ilie Valusescu

ThereforeP2H1

= PH1

andΓ[PH1h, g] = Γ[h,PH1g],

hence PH1 can be interpreted as an ”orthogonal projection” on H1. In the following PH1

will be called the ”Γ-orthogonal projection” of H on H1.Let us remark that the unique element h1 ∈ H, obtained by the ”Γ-orthogonal projec-

tion” of h from H on H1, can belongs not necessary to H1, but sufficiently closed to H1

to be considered as the best estimation, due tu (3.2). This fact will be very usefull in thefollowing, giving us the possibility to use it as an orthogonal projection in determining thepredictable part and the prediction error in estimation, or filtering process.

A Γ-correlated process is a sequence (ft)t∈G in the right L(E)-module H endowed witha correlation of the action of L(E). The set G is Z, R, or more generally a locally compactabelian group. By a Γ-stationary process, or simply a stationary process, we mean asequence ftt∈G from H such that Γ[ft, fs] depends only on the difference s− t and noton s and t separately.

In the particular cases when G = Z, or G = R, the processes are usually called discrete,or continuous processes, respectively.

In this paper we will consider only the case of discrete Γ-stationary processes from H,following to analyze the continuous and nonstationary cases in a separate paper.

The function Γf : Z → L(E) given by

Γf (n) = Γ[f0, fn] (3.4)

is called the correlation function of the process fnn∈Z.Concerning to a Γ-stationary process fnn∈Z ⊂ H , the following prediction submod-

ules are considered in the state space H. The past and present of the process up to themoment n will be the submodule

Hfn =

∑

k

Akfk;Ak ∈ L(E), k ≤ n; (3.5)

the future of the process is

Hfn =

∑

k

Akfk;Ak ∈ L(E), k > n; (3.6)

the remote past

Hf−∞ =

⋂

n

Hfn; (3.7)

and the space generated by the process

Hf∞ =

∑

k

Akfk;Ak ∈ L(E),, (3.8)

the sums being considered for finite actions of L(E) on H.


Also, due the algebraic imbedding h → Vh of H into L(E,K), to the prediction sub-modules from H, the corresponding subspaces in the measuring space K become:the past and present

Kfn =

∨

k≤n

VfkE, or Kfn =

∨

h∈Hfn

VhE; (3.9)

the future

Kfn =

∨

k>n

VfkE, or Kfn =

∨

h∈Hfn

VhE; (3.10)

the remote past

Kf−∞ =

⋂

n

Kfn; (3.11)

and the space generated by the process

Kf∞ =

∞∨

−∞VfkE. (3.12)

Let fn and gn be two Γ-stationary processes in H. We say that fn and gnare stationary cross-correlated processes, if Γ[fn, gm] depends only on the difference m−nand not on m and n separately.

Proposition 3.2. For any stationary processes fnn∈Z in the correlated action E,H,Γthere exists a unitary operator Uf on K

f∞ such that

Vfn = Unf Vf0 . (3.13)

The Γ-stationary process gnn∈Z is stationary cross-correlated with fnn∈Z if and onlyif there exists a unitary operator Ufg on

Kfg∞ = Kf

∞∨

Kg∞,

such thatUf = Ufg|Kf

∞, Ug = Ufg|Kg∞.

Proof. It is enough to define Uf on the generators of Kf∞ by

Uf

(∑

n

Vfnan

)=

∑

n

Vfn+1an. (3.14)

From the fact that∥∥∥∥∥Uf

(∑

n

Vfnan

)∥∥∥∥∥

2

=

∥∥∥∥∥∑

n

Vfn+1an

∥∥∥∥∥

2

=(∑

n

Vfn+1an,∑

k

Vfk+1ak

)=

=∑

n,k

(V ∗fk+1

Vfn+1an, ak

)=

∑

n,k

(Γ[fk+1, fn+1]an, ak

)=

86 Ilie Valusescu

=∑

n,k

(Γ[fk, fn]an, ak

)=

∥∥∥∥∥∑

n

Vfnan

∥∥∥∥∥

2

,

it follows that Uf defines a unitary operator on Kf∞ which verifies (3.13).

Concerning the second part of the Proposition, if fn and gn are stationary cross-correlated processes, then putting

Ufg

(∑

n

Vfnan +∑

m

Vgmbm

)=

∑

n

Vfn+1an +∑

m

Vgm+1bm, (3.15)

we have∥∥∥∥∥Ufg

(∑

n

Vfnan +∑

m

Vgmbm

)∥∥∥∥∥

2

=

∥∥∥∥∥∑

n

Vfn+1an +∑

m

Vgm+1bm

∥∥∥∥∥

2

=

=

∥∥∥∥∥∑

n

Vfn+1an

∥∥∥∥∥

2

+

∥∥∥∥∥∑

m

Vgm+1bm

∥∥∥∥∥

2

+ 2Re∑

n,m

(Γ[gm+1, fn+1]an, bm

)=

=

∥∥∥∥∥∑

n

Vfnan

∥∥∥∥∥

2

+

∥∥∥∥∥∑

m

Vgmbm

∥∥∥∥∥

2

+ 2Re∑

n,m

(Γ[gm, fn]an, bm

)=

=

∥∥∥∥∥∑

n

Vfnan +∑

m

Vgmbm

∥∥∥∥∥

2

.

Hence (3.15) defines a unitary operator Ufg on Kfg∞ which verifies the requested properties,

and the Proposition is proved.

The unitary operator Uf is called the shift operator attached to the Γ-stationary processfn, and Ufg is the extended shift of of the Γ-stationary cross-correlated processes fnand gn.

In what follows, for simplicity, for a Γ-stationary process fn we will write Vf for theoperator Vf0 , and by (3.9) and (3.13)

Kf∞ =

∞∨

−∞Unf VfE. (3.16)

The correlation function n → Γf (n) of a process fn, given by (3.4), is an L(E)-valuedpositive definite function on the group Z. Indeed, for any system of integers (n1, . . . , np)and (a1, . . . , ap) from E

∑

i,j

(Γf (ni − nj)ai, aj

)=

∑

i,j

(Γ[fnj , fni)ai, aj

)=

=∑

i,j

(Γ[fnj , fni)Aia,Aja

)=

(Γ[∑

j

Ajfnj ,∑

i

Aifni

]a, a

)≥ 0,


where again we have used the fact that for (a1, . . . , ap) from E there exists Ai ∈ L(E) anda ∈ E such that Aia = ai.

Analogously, the relation

Γfg(n) = Γ[f0, gn] (3.17)

defines a positive definite L(E)-valued function on the group Z

n → Γfg

which is called the cross-correlation function of the Γ-stationary processes fn and gn.Taking account by (3.4) and(3.13) we have

Γf (n) = Γ[f0, fn] = V ∗f U

nf Vf .

Hence the correlation function n → Γf (n) of a process fn, as an L(E)-valued positive

definite function on the group Z, admits a dilation of the form [Kf∞, Vf , Uf ], and by (3.16)

it follows that this is its minimal unitary dilation.

The triplet [Kf∞, Vf , Uf ] attached to a stationary process fn in the complete cor-

related action E,H,Γ will be called the geometrical model of the process. The nameconsist in the fact that all geometrical elements for the prediction of the process fn canbe found and expressed in the terms of its geometrical model.

For two cross-correlated processes fn and gn, the common geometrical model will

be [Kfg∞ , Vfg, Ufg].

Let us consider now some special processes, which will be necessary in predictionprocedure.

The Γ-stationary process gn ∈ H is a white noise process, if for any n 6= m

Γ[gn, gm] = 0.

The Γ-stationary process fn ∈ H contains the white noise process gn from H if:

(i)fnandgnare cross-correlated andΓ[fn, gm] = 0 for m ≥ n;

(ii)VgE ⊂ Kf0 ;

(iii)ReΓ[fn − gn, gn] ≥ 0.

(3.18)

The Γ-stationary process fn ∈ H is deterministic if and only if does not contain anon-null white noise process.

The Γ-stationary process fn ∈ H is a moving average of a white noise process gn, iffn contains gn and the corresponding generated spaces are the same, i.e. Kg

∞ = Kf∞.

As we will see in the following, in the context of a complete correlated action E,H,Γ,in each stationary process a maximal white noise can be found.

Lemma 3.3. If the Γ-stationary process fn contains the white noise process gn, then

VgnE ⊂ Kfn ⊖K

fn−1. (3.19)

88 Ilie Valusescu

Proof. For each h from Kfn−1 and a, b from E we have

(Vgna, Vhb

)=

(Γ[h, gn]a, b

)=

(Γ[ ∑

k≤n−1

Akfk, gn

]a, b

)=

=∑

k≤n−1

(A∗

kΓ[fk, gn]a, b)= 0,

because Γ[fk, gn] = 0 for k < n, and Lemma is proved.

Beside the factorization theorems, the Wold decompositions are very important insolving the prediction problems. Here aWold type decomposition adequate to the completecorrelated context is presented.

For any stationary process fn in the complete correlated action E,H,Γ, using theΓ-orthogonal projection on a right L(E)-submodule from H given by Proposition 3.1, foreach n ∈ Z we can take

gn = fn − PH

fn−1

fn. (3.20)

As we easily can see, (3.20) defines a Γ-stationary process gn in H. Because gn is the

innovation part of fn with respect to Hfn−1, the process gn will be called the innovation

process of the Γ-stationary process fn.

Theorem 3.4. (Pre-Wold decomposition). The innovation process gn from H associatedwith the Γ-stationary process fn ⊂ H is the maximal white noise process contained infn. For each m < n we have

Kfn = Kf

m ⊕( n⊕

k=m+1

VgkE)

(3.21)

and

Kfn = K

f−∞ ⊕Kg

n. (3.22)

Proof. The innovation process gn associated with fn by (3.20) is a white noise processin H. Indeed, for m 6= n

Γ[gn, gm] = Γ[fn − PH

fn−1

fn, fm − PH

fm−1

fm] =

= V ∗fn(IKf

n− P

Kfn−1

)(IK

fm− P

Kfm−1

)Vfm = 0.

From the definition (3.20) of gn, it is easy to verify that gn is contained in fn,and if g′n is another white noise process contained in fn, then g′n is also containedin gn.

To prove (3.21), let us remark that for the maximal white noise process gn containedin fn, in Lemma 3.3 we have equality

VgnE = Kfn ⊖K

fn−1.


This is equivalent with the fact that

Kfn = K

fn−1 ⊕ VgnE. (3.23)

By an iteration of (3.23) we obtain (3.21). Taking account that Kf−∞ and K

f∞ are

orthogonal subspaces in Kfn, it follows that

Kf−∞ ⊕Kg

n ⊂ Kfn.

For the converse inclusion, let us remark that for k ∈ Kfn we have

PK

f−∞⊕K

gnk = P

Kf−∞

k + PKgnk.

From the fact that Kfn ⊂ K

fn+1, the definition of Kf

−∞ as intersection of Kfn, and because

Kgn =

n∨

−∞VgkE =

n⊕

k=−∞VgkE,

it results that

PK

f−∞⊕K

gnk = lim

m→−∞PK

fmk + lim

m→−∞P n⊕

m+1Vgk

Ek = lim

m→−∞PK

fm⊕

( n⊕m+1

VgkE)k = P

Kfnk = k.

Therefore

Kfn ⊂ K

f−∞ ⊕Kg

n

and it follows that

Kfn = K

f−∞ ⊕Kg

n,

which finish the proof.

Theorem 3.5. (Wold decomposition). The Γ-stationary process fnn∈Z ⊂ H has aunique decomposition of the form

fn = un + vn (3.24)

where unn∈Z ⊂ H is the moving average of the maximal white noise process gnn∈Z ⊂ H

contained in fn, and vnn∈Z ⊂ H is a deterministic process. Also we have

Γ[un, vm] = 0, (n,m ∈ Z). (3.25)

Proof. Using (3.22), for each n ∈ Z we consider

Vfn = PKgnVfn + P

Kf−∞

Vfn . (3.26)

Denoting by

un = PHgnfn and vn = P

Hf−∞

fn, (3.27)

90 Ilie Valusescu

it is easy to see that (3.24) is fulfiled. Taking account by (3.22), the relation (3.25) resultsfrom the fact that for any n,m ∈ Z

Γ[un, vm] = V ∗fnPK

gnPK

f−∞

Vfm .

A process vn ⊂ H is deterministic iff the maximal contained white noise is the nullprocess. Using again (3.22) it results that

Kvn = Kv

−∞. (3.28)

Due to the fact that for any a ∈ E we have

Vfna = Vuna+ Vvna ∈ Kgn ⊕Kv

n,

and it results thatKf

n ⊆ Kgn ⊕Kv

n.

By the fact that Kgn ⊂ K

fn and VvkE ⊂ K

f−∞ ⊂ K

fn, it follows the converse inclusion

Kfn ⊇ K

gn ⊕Kv

n. ThereforeKf

n = Kgn ⊕Kv

n, (3.29)

and from (3.22) it follows that for any n ∈ Z we have Kvn = K

f−∞. It follows that

Kv−∞ =

⋂

n

Kvn = K

f−∞,

and by (3.28) it results that vn is a deterministic process in H.

From the orthogonality of the decomposition Vfn = Vun + Vvn , it follows that Kfn =

Kun ⊕Kv

n, and by (3.29) it results that

Kgn = Ku

n. (3.30)

Moreover, the process gn is contained in un. Indeed, for m ≥ n we have

Γ[un, gm] = Γ[PHgnfn, fm − P

Hfm−1

fm] = Γ[PHgnfn, fm]− Γ[PH

gnfn,PH

fm−1

fm] =

= Γ[PHgnfn,PH

gnfm]− Γ[PH

gnfn,PH

gnPH

fm−1

fm] = 0

andReΓ[un − gn, gn] = ReΓ[fn − vn − gn, gn] =

= ReΓ[fn − gn, gn]− ReΓ[vn, gn] = ReΓ[fn − gn, gn] ≥ 0.

So, taking account by (3.30) the conditions (3.18) are verified, and the process gn iscontained in un. Consequently un is a moving average of gn.

The unicity of the decomposition (3.24) it follows from the unicity of the decomposi-tions

Kf∞ = Ku

∞ ⊕Kv∞ and Kf

∞ = Ku∞ ⊕K

f−∞,

and the proof is finished.


Let us consider a stationary process fnn∈Z in the complete correlated action E,H,Γand fix the present at the moment t = 0. Then its past and present is the right L(E)-submodule of H of the form:

Hf0 =

h ∈ H; h =

∑

k≤0

Akfk, Ak ∈ L(E). (3.31)

The prediction problems for stochastic processes, stationary or non-stationary one, consistin obtaining informations about the considered process at the next moment (in this caset = 1) from the ”knowledge” of the process up to the present moment (t = 0). Wecan obtain informations about the process acting with some specific experiences. In thecomplete correlated case this fact is done by acting with operators from L(E) on the rightL(E)-module H. The results of the experiments are measured in a ”measuring system”intimately related to the nature of the experiments. The ”measuring system” is given bythe metric of the attached measuring space of the correlated action.

Such a way, we can interpret the past and present H0 of the process fn as the totalinformation obtained acting on the process up to the present moment. To predict theprocess at the next step (t = 1) it means to obtain the best information about f1 in terms

of the elements from the fixed past and present Hf0 . Analogously a p-step prediction can

be done, obtaining fp from the informations given by Hf0 .

Theorem 3.6. Let gn be the maximal white noise contained in a Γ-stationary processfn ⊂ H. Setting

f1 = PH

f0f1 = f1 − g1, (3.32)

then Γ[f1, g1] = 0 and

Γ[f1 − f1, f1 − f1] = infh∈Hf

0

Γ[f1 − h, f1 − h] (3.33)

where the infimum is taken in the set of all positive operators from L(E).For any a ∈ E we have

(Γ[f1 − f1, f1 − f1]a, a

)= inf

m∑

j,k=0

(Γ[fj , fk]aj , ak

), (3.34)

where the infimum is taken over all finite systems (a1, . . . , am) of elements in E and a0 = a.

Proof. Taking account that f1 is the Γ-orthogonal projection of f1 on Hf0 , the first part

of the Theorem it results from Proposition 3.1. To prove the second part of the Theorem,for any a ∈ E we have

(Γ[f1 − f1, f1 − f1]a, a

)E=

(Γ[f1 − P

Hf0f1, f1 − P

Hf0f1]a, a

)=

(Γ[g1, g1]a, a

)=

=(Γ[g0, g0]a, a

)=

(V ∗f (I − P

Kf−1)Vfa, a

)=

∥∥∥Vfa− PK

f−1Vfa

∥∥∥2= inf

k∈Kf−1

‖Vfa− k‖2 =

92 Ilie Valusescu

= inf(a1,...,am)∈E

∥∥∥∥∥Vfa+m∑

k=1

U∗kf Vfak

∥∥∥∥∥

2

= inf(a1,...,am)∈E

∥∥∥∥∥m∑

k=0

U∗kf Vfak

∥∥∥∥∥

2

=

= inf(a1,...,am)∈E

( m∑

j=0

U∗jf Vfaj ,

m∑

k=0

U∗kf Vfak

)= inf

(a1,...,am)∈E

m∑

j,k=0

(Uk−jVfajVfak

)=

= inf(a1,...,am)∈E

m∑

j,k=0

(UkVfaj , U

jVfak)= inf

(a1,...,am)∈E

m∑

j,k=0

(Vfkaj , Vfjak

)=

= inf(a1,...,am)∈E

m∑

j,k=0

(V ∗fjVfkaj , ak

)= inf

(a1,...,am)∈E

m∑

j,k=0

(Γ[fj , fk]aj , ak

).

Thus (3.34) is proved, and the proof is finished.

From the above theorem we can see that, if in some way we can determine f1, then itcontains the best information about f1 which can be extract acting on the process up tothe moment t = 0. This is the reason why f1 is called the predictable part of f1 and theoperator

∆[f ] = Γ[f1 − f1, f1 − f1] (3.35)

is called the prediction-error operator.

Concerning the prediction-error operator, let us remark that (3.34) is an intrinsicformula of computing the prediction-error in terms of the actions and of the correla-tions. Later, after attaching a “maximal function” to each Γ-stationary process, thenthe prediction-error operator will be obtained in terms of the coefficients of its maximalfunction.

Let us remark that

0 ≤ ∆[f ] ≤ Γf (0). (3.36)

If in (3.36) equality holds, then 0 = ∆[f ] = Γ[f1 − f1, f1 − f1] = Γ[gn, gn], where gnis the maximal white noise contained in fn. Therefore gn = 0 and it results that fncontains no non-null white noise, consequently fn is a deterministic process.

Conversely, if fn is deterministic, then its maximal white noise is null and it followsthat ∆[f ] = Γ[gn, gn] = 0.

Moreover, if fn is deterministic, then, because gn = 0 it results that fn = fn, whichis equivalent with the fact that P

Kfn−1

Vfn = Vfn , and it follows that for each n ∈ Z we have

Kfn−1 = K

fn. Hence the process fn ⊂ H is deterministic if and only if Kf

−∞ = Kfn = K

f∞.

So we have the following

Lemma 3.7. The following are equivalent:

(i) The process fn ⊂ H is deterministic;

(ii) ∆[f ] = 0;

(iii) Kf−∞ = K

fn = K

f∞.


If in (3.36) we have ∆[f ] = Γf (0), then fn is a white noise process. Indeed, letn 6= m, or for convenience n < m . Using (3.33) it results that for any ε > 0

(Γf (0)a, a

)=

(∆[f ]a, a

)≤

(Γ[fm ± εfn, fm ± εfn]a, a

)=

=(Γf (0)a, a

)+ ε2

(Γf (0)a, a

)± 2εRe

(Γ[fm, fn]a, a

).

Therefore for any ε > 0 we have ε(Γf (0)a, a

)≥ ±2Re

(Γ[fm, fn]a, a

)and it follows that

Re(Γ[fm, fn]a, a

)= 0 for any a ∈ E.

Analogous it is obtained that Im(Γ[fm, fn]a, a

)= 0, and it results that for n 6= m we

have Γ[fm, fn] = 0, i.e. fn is a white noise process.Conversely, if fn is a white noise process, then its maximal white noise is itself, and

∆[f ] = Γ[fn, fn] = Γf (0).

So we have proved the following

Lemma 3.8. The Γ stationary process fn ⊂ H is a white noise process if and only if∆[f ] = Γf (0).

Corollary 3.9. If gn is the maximal white noise process contained in the Γ-stationaryprocess fn, then

∆[f ] = ∆[g]. (3.37)

Generally, the prediction-error operator of the maximal white noise gn contained infn will be less that the prediction-error operator of fn, as will be seen in the followingProposition.

Proposition 3.10. If gn is the maximal white noise contained in the Γ-stationary pro-cess fn, then

∆[g] ≤ ∆[f ].

Proof. Because gn is contained in fn, from (3.18) it follows that for n < 0 we have

Γ[fn, g0] = 0. Consequently Γ[h, g0] = 0 for any h ∈ Hf−1. Also we have

Γ[f0 − h, f0 − h] = Γ[f0 − (h+ g0) + g0, f0 − (h+ g0) + g0] =

= Γ[f0 − (h+ g0), f0 − (h+ g0)] + Γ[g0, g0] + 2ReΓ[f0 − (h+ g0), g0] =

= Γ[f0 − (h+ g0), f0 − (h+ g0)] + Γ[g0, g0] + 2ReΓ[f0 − g0, g0].

Using again the fact that gn is contained in fn we have that ReΓ[f0 − g0, g0] ≥ 0,

and from the previous relations it results that Γ[g0, g0] ≤ Γ[f0−h, f0−h] for any h ∈ Hfn−1.

Applying now Lemma 3.8 it follows that ∆[g] ≤ Γ[f0 − h, f0 − h] for any h ∈ Hfn−1,

and by Theorem 3.6 it results that ∆[g] ≤ ∆[f ], and the proof is finished.

Proposition 3.11. The Γ-stationary process fn is a moving average of a white noisegn, if and only if

Kf−∞ = 0. (3.38)

94 Ilie Valusescu

Proof. Let [Kf∞, Vf , Uf ] be the geometrical model of the Γ-stationary process fn and

[Kg∞, Vg, Ug] the geometrical model of the white noise gn contained in fn. As we have

seen Kfn ⊂ K

fn+1, and it follows that

Kf∞ = K

f−∞ ⊕

[ +∞⊕

−∞(Kf

n ⊖Kfn−1)

]. (3.39)

If fn is a moving average of gn, then taking account by Lemma 3.3 we have

Kf∞ = Kg

∞ =+∞∨

−∞VgnE ⊆

+∞⊕

−∞(Kf

n ⊖Kfn−1) ⊆ Kf

∞

and it results that Kf−∞ = 0.

Conversely, if gn is the maximal white noise process contained in fn and Kf−∞ =

0, then

Kg∞ =

+∞⊕

−∞(Kf

n ⊖Kfn−1) = K

f−∞ ⊕

[ +∞⊕

−∞(Kf

n ⊖Kfn−1)

]= Kf

∞,

and it follows that fn is a moving average of its maximal white noise process gn.

Let us remark that the previous results are obtained in the context of a completecorrelated action E,H,Γ. If E,H,Γ is not complete, but only a correlated action,then [14] it can happen that a stationary process to contain no white noise, without to bea deterministic process. In this case it is valid only the affirmation that, if the process isdeterministic, then it contain no white noise. Also we can have K

f−∞ = 0 without to

be a moving average of contained white noises. These things are happening because inthe context of a non-complete correlated action we can not control whole range of whitenoises, as we can do in the complete correlated case, by the maximal white noise.

The main prediction problem which arise in the context of a correlated action, or acomplete correlated action E,H,Γ, is to obtain a linear predictor, i.e. to determine asequence of operators Ak in L(E) such that

∑k

AkVf−kto be strongly convergent to Vf1

in L(E,K). Generally, this problem is rather difficult, but in the context of a completecorrelated action, under some supplimentary condition, similar with the condition imposeby Wiener and Masani [21] in the matricial case, the predictable part fn of the processfn can be obtained. This will be done later, after obtaining some more informations asconsequences of the investigations made on the process.

Let [Kf∞, Vf , Uf ] be the geometrical model attached to a stationary process fn in

the complete correlated action E,H,Γ, and n → Γf (n) be the corresponding correlationfunction.

By Naimark dilation theorem, there exists a unique L(E)-valued semispectral measureF on T such that

Γf (n) =

∫ 2π

0e−intdF (t). (3.40)


This L(E)-valued semispectral measure, uniquelly attached to the stationary process fnfrom H, is called the spectral distribution of the process.

Let [K, V, E] be the minimal spectral dilation of the L(E)-valued semispectral measureF , and U be the unitary operator corresponding to the L(K)-valued spectral measure E.

It is not hard to see that [K, V, U ] coincides with [Kf∞, Vf , U

∗f ] as a minimal unitary

dilation of the L(E)-valued positive definite function n → Γf (n) on the group Z.Let E,F,Θ(λ) be the maximal function of the spectral distribution F of the process

fn. This maximal outer L2-bounded function will be also called the maximal function ofthe process fn, and will play an important role in the prediction of the process. Actuallythe predictable part (linear predictor) and the prediction-error operator will be obtainedin terms of the coefficients of the maximal function of the process.

From the factorization Theorem 2.2 and Proposition 3.11 it results that the spectraldistribution of the process fn is an L(E)-valued semispectral measure of analytic typeand coincides with the L(E)-valued semispectral measure attached to the maximal function

F +FΘ, if and only if Kf−∞ = 0, i.e., if and only if the process fn is a moving average

of a white noise.

Let ∆[F ] be the Szego operator attached to F as in (2.16). Then we have the followingresult.

Theorem 3.12. If fn ⊂ H is a Γ-stationary process, ∆[f ] is the prediction-error op-erator, and ∆[F ] is the Szego operator attached to the spectral distribution of the processfn, then

∆[f ] = ∆[F ]. (3.41)

Proof. Using (3.34) and (2.16), fo any a ∈ E we have

(∆[f ]a, a

)= inf

m∑

j,k=0

(Γ[fj , fk]aj , ak

)= inf

m∑

j,k=0

(Γf (k − j)aj , ak

)=

= inf

m∑

j,k=0

∫ 2π

0ei(j−k)td

(F (t)aj , ak

)=

(∆[F ]a, a

),

where infimum is taken over all systems of finite elements (a1, . . . , am) ⊂ E, such thata0 = a.

Therefore ∆[f ] = ∆[F ], and the proof is finished.

From (3.41) and (2.19) it results another estimation of the prediction-error operator,in terms of the maximal function of the process, namely

∆[f ] = Θ(0)∗Θ(0). (3.42)

In the following theorem a spectral characterization of the Wold decomposition is given.

Theorem 3.13. Let fn be a stationary process in the complete correlated action E,H,Γ,∆[f ] be its prediction-error operator and fn = un + vn be the Wold decomposition of the

96 Ilie Valusescu

process. If F is the spectral distribution of the process fn and E,F,Θ(λ) the maximalfunction of the process, then:

(i) FΘ is the spectral distribution of the moving average part un of the process fn.(ii) ∆[f ] = Θ(0)∗Θ(0) and dim∆[f ]E = dimF.

Moreover, the process fn is deterministic if and only if there exist no non-nul L2-bounded analytic function E,F′,Θ′(λ) such that FΘ′ ≤ F . The process fn is a movingaverage, if and only if FΘ = F

Proof. Using the previous proved coincidence of [K, V, U ] and [Kf∞, Vf , U

∗f ], taking account

of the way as the maximal function was found in Theorem 2.2 and un in the Theorem 3.5,it follows the assertion (i). The assertion (ii) it results from Theorem 3.12 and Theorem2.4

This way we obtain a characterization of the Wold decomposition in terms of factora-bility of the spectral distribution.

As was already mentioned, the main prediction problem is to obtain the linear pre-dictor, i.e. to determine a sequence of operators Ak in L(E) such that

∑k AkVf−k

tobe strongly convergent to the predictable part Vf1

in L(E,K). This means to obtain the

predictable part of f1 in terms of the fixed past and present Hf0 .

Under a boundedness condition of Harnack type on the spectral distribution of theprocess fn, the predictable part fn will be obtained using a linear filter consisting insuccesive actions on the process up to the present moment. The coefficients of the filter willbe determined in terms of the Taylor coefficients of the maximal function of the processand its inverse.

The boundedness condition impossed to the spectral distribution F of the process isthe Harnack equivalence with the normalized Lebesgue measure, i.e. there exists a strictpositive constant c such that

1

2πc dt ≤ F ≤ 1

2πc−1dt. (3.43)

Theorem 3.14. Let F be an L(E)-valued semispectral measure on T, having the maximalfunction E,F,Θ(λ), and G = Θ(0)∗Θ(0). Then F verifies the condition (3.43) if andonly if FΘ = F , dimE = dimF, E,F,Θ(λ) is a bounded analytic function which hasa bounded inverse, and there exists an identification of E,F,Θ(λ) with an invertiblebounded analytic function E,E,Φ(λ) such that

Φ(0) = G1/2. (3.44)

Proof. Let E,E,Φ(λ) be an identification of E,F,Θ(λ) with the requested properties,and E,E,Ψ(λ) be its inverse. The functions Φ(λ) and Ψ(λ) being bounded analytic onD, there exists the Fatou limits Φ(eit) and Ψ(eit) a.e. on T and

dF = dFΦ =1

2πΦ(eit)∗Φ(eit)dt. (3.45)


For any trigonometric polynomial p and a ∈ E we have

∫ 2π

0

∣∣p(eit)∣∣2 d

(F (t)a, a

)=

1

2π

∫ 2π

0

∥∥Φ(eit)p(eit)a∥∥2 dt ≤ ‖Φ‖2 1

2π

∫ 2π

0

∣∣p(eit)∣∣2 ‖a‖2 dt

and also ∫ 2π

0

∣∣p(eit)∣∣2 d

(F (t)a, a

)=

1

2π

∫ 2π

0

∥∥Φ(eit)p(eit)a∥∥2 dt ≥

≥ ‖Ψ‖−2 1

2π

∫ 2π

0

∥∥Ψ(eit)Φ(eit)p(eit)a∥∥2 dt = 1

2π‖Ψ‖−2

∫ 2π

0

∣∣p(eit)a∣∣2 dt · ‖a‖2 ,

where Φ and Ψ are the linear bounded operators attached to Φ(λ), respectively Ψ(λ). Itfollows that for any positive continuous function ϕ on T we have

1

2π‖Ψ‖−2

∫ 2π

0ϕdt ≤

∫ 2π

0ϕdF ≤ 1

2π‖Φ‖2

∫ 2π

0ϕdt

and it results that the spectral distribution F verifies (3.43).

Conversely, suppose that the semispectral measure F verifies (3.43). If [K, V, E] is thespectral dilation of F , and U is the unitary operator on K corresponding to E, then

X1

(∑

n

UnV an

)=

∑

n

eintan

defines an invertible operator from K+ into H2(E) which intertwines U with the shiftoperator on H2(E). Then clearly

X1

( ⋂

n≥0

UnK+

)=

⋂

n≥0

eintX1K1 ⊆⋂

n≥0

eintH2(E) = 0.

It results that ⋂

n≥0

UnK+ = 0

and by Theorem 2.2 it follows that FΘ = F .

From (3.43) it results that E,F,Θ(λ) is bounded and the attached operator Θ+

from H2(E) into H2(F) is bounded, with a bounded inverse Θ−1+ . Since Θ−1

+ intertwinesthe shifts on H2(F) and H2(E), it follows that there exists a bounded analytic functionF,E,Ω(λ) which is the inverse of E,F,Θ(λ).

Let us consider the operator X : F → E defined by

X +G1/2Ω(0), (3.46)

where G = Θ(0)∗Θ(0). Then for any b ∈ F it results that

‖Xb‖2 =∥∥∥G1/2Ω(0)b

∥∥∥2=

(GΩ(0)b,Ω(0)b

)=

(Θ(0)∗Θ(0)Ω(0)b,Ω(0)b

)= ‖Θ(0)Ω(0)b‖2 = ‖b‖2 .

98 Ilie Valusescu

Hence (3.46) defines a unitary operator X from F on E. If we put

Φ(λ) = XΘ(λ) (λ ∈ D), (3.47)

then we have

Φ(0) = XΘ(0) = G1/2Ω(0)Θ(0) = G1/2,

and it results that (3.47) defines a bounded analytic function E,E,Φ(λ) with a boundedinverse, which coincides with E,F,Θ(λ), satisfying (3.44), and the proof is finished.

Now, let us consider fn be a stationary process in the complete correlated actionE,H,Γ, whose spectral distribution F satisfies the condition (3.43). Then its prediction-error operator ∆[f ] = G is an invertible operator on E. Let gn be the maximal whitenoise contained in fn. If we define the process hn given by

hn = G−1/2gn, (3.48)

then it is easy to see that hn is a white noise process with the property

Γ[hn, hn] = IE. (3.49)

The process hn given by (3.48) is called the normalized innovation process of theΓ-stationary process fn.

Due to the coincidence between E,F,Θ(λ) and E,E,Φ(λ), in the following the max-imal function of the process fn will be denoted by E,E,Θ(λ). Then, the geometricalmodel for prediction can be drown as follows:

Kf∞ = L2(E), K+ = L2

+(E)

V = Θ|E, (V a)(t) = Θ(eit)a,

and Uf is the operator of multiplying by e−it on L2(E).

Therefore we can identify the considered processes fn, gn and hn from H withthe following operators from E into L2(E):

fn : a −→ e−intΘ(eit)a

gn : a −→ e−intΘ(0)a = e−intG1/2a

hn : a −→ e−inta.

Also the Taylor expansions of the maximal function E,E,Θ(λ) and its inverse E,E,Ω(λ)can be written as

Θ(λ) = G1/2 +∞∑

k=1

Θkλk (λ ∈ D), (3.50)

Ω(λ) = G−1/2 +

∞∑

k=1

Ωkλk (λ ∈ D). (3.51)


Theorem 3.15. Let fn be a Γ-stationary process whose spectral distribution F verifiesthe boundedness condition (3.43). Then we have

fn =∞∑

k=0

Θkhn−k (3.52)

and

hn =∞∑

k=0

Ωkfn−k, (3.53)

where the series are supposed to be convergent in the strong topology on L(E,K).

Proof. Working with the above identifications of the processes fn, gn and hn wehave for any a ∈ E

∞∑

k=0

Θkhn−ka =∞∑

k=0

e−i(n−k)tΘka = e−int∞∑

k=0

e−iktΘka = e−intΘ(eit)a = fna

and ∞∑

k=0

Ωkfn−ka =∞∑

k=0

e−i(n−k)tΘ(eit)Ωka = e−intΘ(eit)∞∑

k=0

eiktΩka =

= e−intΘ(eit)Ω(eit)a = e−inta = hna,

where the convergence of the series and the commutation of operators with the abovesummation are verifiable in an obvious manner.

Theorem 3.16. Let fn be a Γ-stationary process whose spectral distribution F veri-fies the boundedness condition (3.43), E,E,Θ(λ) be the maximal function of fn andE,E,Ω(λ) be its inverse. Then the predictable part fn of fn is given by

fn =∞∑

j=0

Ejf(n−1)−j , (3.54)

where Ej are given by

Ej =

j∑

p=0

Ωj−pΘp+1. (3.55)

The predicton-error operator ∆[f ] is

∆[f ] = Θ(0)∗Θ(0). (3.56)

Proof. The relation (3.46) actually summarize the result obtained about the prediction-error operator in (3.42). Concerning the other part, taking account on the previous theo-rem, it results that

fn = fn−gn =∞∑

k=0

Θkhn−k−G1/2hn =∞∑

k=1

Θkhn−k =∞∑

k=1

Vhn−kΘk =

∞∑

k=1

V∑sΩsfn−k−s

Θk =

100 Ilie Valusescu

=∞∑

k=1

∞∑

s=0

Vfn−k−s(ΩsΘk) =

∞∑

k=1

∞∑

s=0

ΩsΘkfn−k−s =∞∑

p=0

∞∑

s=0

ΩsΘp+1f(n−1)−(p+s) =

=∞∑

j=0

( ∑

p+s=j

ΩsΘp+1

)f(n−1)−j =

∞∑

j=0

( j∑

p=0

Ωj−pΘp+1

)f(n−1)−j =

∞∑

j=0

Ejf(n−1)−j .

Therefore the predictable part fn of fn can be obtained using the linear (infinite) filterE1, E2, . . ., so called the linear predictor, or the Wiener filter for prediction.

As previously was mentioned, this paper presents a way to solve the prediction prob-lems in the infinite-dimensional case, but in the meantime other methods to investigateΓ-correlated processes was found. Using some structure results from operatorial extrapo-lation theory (see e.g. [9]), a formula for the predictable part fn of fn can be obtained interms of the so called choice sequence attached to the correlation function of the process.Using choice sequence techniques, a solution for the finite past prediction is presented in[17] and a linear filter for its predictable part is obtained into the form

fn+1 =n∑

j=0

Ejfn−j .

Here the discrete time Γ-correlated processes was analysed, but the same can be done,with specific operator tools, for the continuous case ftt∈R. An interesting study can befound for two-parameters processes, t = (t1, t2) ∈ Z2, where more possibilities to choosethe past arise, and the role of the maximal function attached to a semispectral measureis played by a couple of operator valued functions on the bidisc [3]

Θ = [E,E+; Θ+, E,E−; Θ−]

so called a Wold-type function on D2. The function E,E+,Θ+ is called the analytic partof the Wold-type function Θ, and the function E,E−,Θ− is the modified analytic part ofthe Wold-type function Θ. The name is due to the fact that this function corresponds toan operator valued analytic function on (D×∆) ∪ D2 ∪ (∆× D), where ∆ = C\D. Moreabout these and for some classes of nonstationary processes, especially for periodicallyΓ-correlated processes will be done in a separate paper.

References

[1] A. DEVINATZ, The factorization of operator valued functions, Ann. of Math., 73(1961), 458-495.

[2] R.G. DOUGLAS, On factoring positive operator functions, J. Math. and Mech., 16(1966), 119-126.


[3] D. GASPAR, N. SUCIU and I.VALUSESCU, Semispectral measures on the bitorusand Wold-type functions on the bidisc, Rev. Roum. Math. Pures et Appl., 38 (1993),227-241.

[4] H. HELSON and D. LOWDENSLAGER, Prediction theory and Fourier series anseveral variables I. and II, Acta Math., 99 (1958), 165-202, and 106 (1961), 175-213.

[5] A.Ya. KHINTCHINE, Korrelationstheorie der stationaren stochastichen Prozessen,Math. An., 109 (1934), 605-615.

[6] A.N. KOLMOGOROV, Sur l’interpolation and extrapolation des scietes stationaires, Comt. Rend. Acad. Sci. Paris, 208 (1939), 2043-2045.

[7] D.B. LOWDENSLAGER, On factoring matrix valued functions, Ann. of Math. 78(1963), 450-454.

[8] H. SALEHI, The continuation of Wiener’s work on q-variate linear prediction and itsextension to infinite-dimensional spaces, Norbert Wiener Collected Works, Vol.III,(ed. P.Masani), MIT Press Cambridge, London, 1981, 307-337.

[9] I. SUCIU, Operatorial extrapolations and prediction, Special classes of linear operatorsand other topics. Birkhauser Verlag, Basel, 1988, 291-301.

[10] I. SUCIU and D. TIMOTIN, On the notion of completeness in prediction theory,Prediction Theory and Harmonic Analysis, The P.Masani Volume (V.Mandrekar andH.Salehi eds.) North-Holland Publ. Co., 1983, 367-378.

[11] I.SUCIU and I. VALUSESCU, Factorization of semispectral measures, Rev. RoumaineMath. Pures et Appl. 21, 6 (1976), 773-793.

[12] I.SUCIU and I. VALUSESCU, Factorization theorems and prediction theory, Rev.Roumaine Math. Pures et Appl. 23, 9 (1978), 1393-1423.

[13] I.SUCIU and I. VALUSESCU, Fatou and Szego theorems for operator valued func-tions, Proc. Roumanian-Finish Sem. 1976, Springer Lect. Notes in Math. 743 (1979),656-673.

[14] I.SUCIU and I. VALUSESCU, A linear filtering problem in complete correlated ac-tions, Journal of Multivariate Analysis, 9, 4 (1979), 559-613.

[15] G. SZEGO, Uber die Randwerte analytischer Functionen, Math. Ann. 84 (1921),232-244.

[16] B. Sz.-NAGY and C. FOIAS, Harmonic analysis of operators on Hilbert space, AcadKiado, Budapest, North Holland Co., 1970.

[17] I. VALUSESCU, Stationary processes in complete correlated actions, Mon. Math 80,2007, Universitatea de Vest Timisoara.

102 Ilie Valusescu

[18] I. VALUSESCU, Stochastic processes in correlated actions, Mon. Math 83, 2008,Universitatea de Vest Timisoara.

[19] V.N. ZASUHIN, Theory of multivariate stationary processes, C.R. (Doklady) Acad.Sci. URSS 33 (1941), 435-437.

[20] N. WIENER, The extrapolation, interpolation and smoothing of stationary time se-ries, New York, 1950.

[21] N. WIENER and P. MASANI, The prediction theory of multivariate stochastic pro-cesses I. and II., Acta Math. 98 (1957), 111-150, and Acta Math. 99 (1958), 93-139.

[22] H. WOLD, A study in the analysis of stationary time series, Uppsala, 1938, 2-nd ed.Stockholm, 1954.

Proceedings of the First Conference „Classical and Functional Analysis”, Azuga – România, September 28-29, 2013, Editor: Nicolae Tița

OLD AND NEW IN THE MONOGENIC QUATERNION THEORY

Dan Pascali *)

Dedicated to the memory of Acad. Prof. Grigore C. Moisil (1906-1973)

Abstract The purpose of this paper is two-fold. First part makes a survey of some basic elements and representations of monogenic functions of one quaternionic variable, while in the second part we apply the theoretical results to the inversion of a singular integral equation in three-dimensional spaces.

AMS Subject Classification (2000): 30E20,30G35, 30G65, 45E05 Key words: Hypercomplex functions, Areolar primitive representations,

Quaternionic Riemann-Hilbert problems

Extensions of the formalism of functions of one complex variable to higher dimensional spaces have dated from the early 1930’s. The “founding fathers of spatial monogenic functions” are Grigore C. Moisil [33]-[34], in 1931, and Rudolf Fueter [16], some years later. At present, there is a huge interest for the investigation of functions of hypercomplex variables, mainly in the framework of Clifford algebras. Unlike the historical survey on quaternions due to H. Malonek [29] in 2003 and the recent monograph of G. Gentili and others [18] in 2013, we emphasize G. C.Moisil’s prior publications in 1931. Moreover, in the same year, for extensions to the three-dimensional space, in the joint work of Gr.C. Moisil- N.V. Teodorescu [37], a matrix procedure has been used.

The present work intends a revaluation of author’s research [42],[43] in the 1960’s, on quaternionic function analysis, according to the actual trends of the Clifford analysis. The mentioned papers were published in Romanian and Russian, with a limited out-spread. For the sake of simplicity, we have avoided of the exterior calculus presentation, remaining in favour with Moisil’s original and direct approach, in which the noncommutativity appears more transparent. The purpose is a close form of the solution in the inversion of A.V. Bicadze’s system of singular integral equations [11], which put into practice the monogenic quaternionic. This method was subsequently augmented by V.I. Shevcenko [57], S. Bernstein [8] and R. Abreu - J. Bory [3]. *) Courant Institute, New York University e-mail address: [email protected]

DAN PASCALI 104

§1. Areolar derivative

Loosely speaking, the equivalence between holomorphy and monogeneity has stimulated the great interest for the function theory of one complex variable .z x iy= + Let :f D be a function, ( ) ( , ) ( , )f z u x y iv x y= + , where D is a domain in . A function ( )f z is holo-morphic if it can be developed as a convergent power series in a neighborhood of each point of its domain of definition ,D while ( )f z is monogenic if its real part ( , )u x y and imaginary part ( , )v x y satisfy the Cauchy-Riemann equations for .x iy+ ∈D Further below,

we substitute the Cauchy-Riemann system by the equation .z f o∂ = Indeed, since it has been

introduced by D. Pompeiu [51] in 1912, the areolar derivative

z

f 1 f f 1 u v u vf i i

z 2 x y 2 x y y x

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ = = + = − + +

∂ ∂ ∂ ∂ ∂ ∂ ∂

shows to be a functional measuring the deviation from monogeneity of such a general function .f Moreover, for a complex function ,f with continuous areolar derivative, the integral representation

(1) ( ) ,( )

( )0 ,

f z for z1 f fd T z

for z2 i z z

ζζ

π ζΓ

∈∂ + = ∈− ∂ ∫

D

D

called the Cauchy-Pompeiu representation, where

(2) ( )

, ,( )g1

d d iTg zz

ζξ η ζ ξ η

ζπ= += −

−∫∫D

is the Pompeiu operator,D being the domain interior to a rectificable curve Γ in the complex plane . Due to this equivalence, the equation z f o∂ = is called also the Cauchy-

Riemann system. In his report to the Fifth International Congress of the Mathematicians in 1912, É. Borel [12] has confirmed that the crucial formula (1) is due basically to D. Pompeiu. Moreover, the first properties of the areolar primitive (2) were emphasized by D. Pompeiu in several papers including in [52]. In the frame of polygene functions (derivable in two variables), the areolare derivative theory was developed in the thesis of M. Nicolescu and Gh. Călugăreanu at Sorbonne in 1928. The American scientists like P.V. Ketchum, E. Kasner, R.N. Haskell, J.A. Ward, V.C. Poor or great Russian scholars A.V. Bicadze, I.N. Vekua and others mentioned the Romanian origins of the areolar derivative.

A noticeable subsequent extension of the representation (2) to functions ( )of C∈ D with

the continuous areolar derivative inD has been developed N.Teodorescu [61]-[62]. Later in 1959, I.N.Vekua [63] took into account the weak derivative z∂ (in the sense of distributions

theory) and studied the behavior of the integral operator (2) on Sobolev spaces. The first extensions of the areolar derivative theory to the three-dimensional space have

been revealed by Romanian mathematicians in the early 1930’s. Two procedures were initially distinguished, namely, Gr. C. Moisil [33]-[34] employed the quaternions while in the joint work [37] with N. V. Teodorescu made use of a system of Dirac square matrices of the fourth order. Both procedures were generalized subsequently the higher-dimensional spaces.

OLD AND NEW MONOGENIC QUATERNION THEORY

105

The matrix method was surveyed in the author’s papers [46]-[47] while the hypercomplex function theory has been presented in the well-known monograph of V. Iftimie [23].

§2. Monogenic quaternions We denote by H the non-commutative associative algebra of real quaternions defined by

, , , , o o 1 1 2 2 3 3 jq q e q e q e q e q j o 1 2 3= = + + + ∈ =H

where oe is the unit element of the multiplication and together with other three basic units

, , ,je j 1 2 3,= satisfies the axioms:

(4)

21, ,

, , ,

, , , .

2 2o 2 3

1 2 3 2 3 1 3 1 2

o k k j k k j

e 1 e e e 1

e e e e e e e e e

e e e e e e e j k 1 2 3

= = = = −

= = = = = − ∀ ≠ =

To an element o o 1 1 2 2 3 3q q e q e q e q e= + + + ∈H there corresponds the conjugate element

o o 1 1 2 2 3 3q q e q e q e q e= − − − ∈H and its norm

.2 2 2 2o 1 2 3q q q q q q q= ⋅ = + + +

Moreover, every nonzero element q ∈H has a multiplicative inverse and .21q q q

−− =

Hence H is a division ring1). Quaternionic analysis can be seen as a natural generalization of the methods of complex analysis to three and four dimensions. Further on, we are interested in quaternionic-valued functions defined on a three-dimensional domain .Ω To a fix point ( , , )1 2 3P P p p p= in

Ω there corresponds the quaternion .1 1 2 3 3p p e p e p e= + +

Let us introduce the three-dimensional Nabla-operator 2

1 2 31 3

e e ex x x

∂ ∂ ∂∇ = + +

∂ ∂ ∂

for maps : .3Φ H To distinguish between Φ ∈ker∇ such that 0∇ ⋅Φ = or 0 ,= Φ ⋅∇ due to noncomutativity, we will consider the left and right operators

3

1 2 3 i1 2 3 ii 1

D e e e ex x x x=

∂ ∂ ∂ ∂= + + =

∂ ∂ ∂ ∂∑ and

3

1 2 3 i1 2 3 ii 1

D e e e ex x x x=

∂ ∂ ∂ ∂= + + =

∂ ∂ ∂ ∂∑ ,

where , , ,1 2 31 e e e are the quaternionic units To a generic point 3( , , )1 2 3Q Q x x x= ∈ there

corresponds the quaternionic variable 1 1 2 3 3q x e x e x e= + + and we will write q ∈ Ω if

( , , ) .1 2 3x x x ∈Ω The quaternionic-valued function ( , ),1CΦ ∈ Ω H namely,

3( ) ( , , ) ( , , ) ( , , ) ( , , )o 1 2 3 1 1 2 3 1 2 1 2 3 2 1 2 3 3q x x x x x x e x x x e x x x eΦ = Φ + Φ + Φ + Φ ,

1 ) A field is a commutative ring that contains the multiplicative inverse for any nonzero element. A ring in which division is possible, but commutativity is not assumed is called skew field or division ring.

DAN PASCALI 106

with ( ), , , , ,1j C j o 1 2 3Φ ∈ Ω = is called monogenic in Ω if it satisfies the equation ( ) 0D qΦ =

for all ,q ∈Ω while ( )qΦ is co-monogenic in Ω if it satisfies the equation ( ) 0D qΦ = for all .q ∈ Ω In a more detailed form, the quaternion ( )qΦ is monogenic if

( ) + + + 2

3 3

3 o 31 21

1 2 1 2

D q ex x x x x x

∂Φ∂Φ ∂Φ ∂Φ∂Φ ∂Φ Φ = − + − + ∂ ∂ ∂ ∂ ∂ ∂

+ + 01 2

o 3 o1 2 12 3

2 3 3 1

e ex x x x x x

∂Φ ∂Φ ∂Φ∂Φ ∂Φ ∂Φ+ − + − =

∂ ∂ ∂ ∂ ∂ ∂

holds for all ( , , ) .1 2 3x x x ∈Ω . Similarly, ( )qΦ is co-monogenic if

( ) + + + 3

3

3 o1 2 21

1 2 1 3 2

D q ex x x x x x

∂Φ ∂Φ ∂Φ∂Φ ∂Φ ∂Φ Φ = − + − + ∂ ∂ ∂ ∂ ∂ ∂

+ + 0o 3 o1 1 22 3

2 1 3 3 2 1

e ex x x x x x

∂Φ ∂Φ ∂Φ∂Φ ∂Φ ∂Φ+ − + − =

∂ ∂ ∂ ∂ ∂ ∂ .

The equality D DΦ = − Φ corresponds (leaving aside the factor )12i

to the relation

, ( ),f f

f u ivz z

∂ ∂= = +

∂ ∂ in the plane, i.e., the quaternion Φ is monogenic in Ω if and only if

Φ is co-monogenic in .Ω Indeed, 0D DΦ = Φ = in .Ω Moreover, by iteration of the above

operators, ( )23D D DΦ = Φ = −∆ Φ , ( ) ,2

3D D DΦ = Φ = −∆ Φ where 3∆ denotes the three-

dimensional Laplacian, and so the ( , )2C Ω H -components of a monogenic or co-monogenic

quarternion in Ω are harmonic functions in .Ω

The prototypical example of a Clifford algebra is n equipped with the standard

Euclidean metric. The latter is the associated algebra nA freely generated over real numbers

by , ... , ,1 ne e the canonical orthonormal basis ,n subjected to the constraints

, ,2je 1 1 j n= − ≤ ≤ and , .j k k je e e e 1 j k n= − ≤ ≠ ≤

For example, oA simply reduces to , the field of real numbers, 1A is , the commutative

field of complex numbers, while 2A becomes H , the skew field of quaternions.

There is a natural embedding nn A given by ( ,..., )1 n j jx x x e∑ and an

associated Dirac operator .n

jjj 1

D ex

=

∂=

∂∑ Starting from this, a rich theory of Clifford

monogenic functions, i.e., functions annihilated by ,D can be developed. Generalizations of the Cauchy-Pompeiu representation to finite-dimensional associative algebras were established by H. Snyder [58] and, a more general frame, by M.N. Roșculeț [52]. In the case of quaternionic monogenic functions, Le-de Huan [27] derived extensions like Liouville’theorem and the modulus theorem.


107

Among all associative algebras over , only the quaternions H share with the property of being a division algebra. It is reasonable to expect that a theory of regular functions :Φ H H might yield results comparable in depth to those of the complex functions. A classical selfcontained account of the main line connections, by using the exterior differential calculus, is due to A. Sudbery [60]. In a recent monograph G. Gentili, C. Stoppato and D.C. Struppa [18] enlarged the theory of polynomials and power series with quaternionic coefficients as well as the new treatment of slice regular functions of one quaternionic variable. An introductory text on the topic of Clifford analysis is presented by J. Ryan [54]. A passing from the functions theory of one complex variable to the quaternions and Clifford algebras or hypercomplex analysis is gradually developed in [21].

Finally, an analogue symbolism to study the monogenic functions was introduced by M.S. Șneerson [58]. Let Ω be a three-dimensional domain, p a scalar function and

( , , )1 2 3a a a a=

a vector function. Suppose that , jp a are continuously derivable functions

in .Ω The couple ( ; )p a

is said to be monogenic in Ω provided

(5) 0,

o.

grad p rot a

div a

+ =

=

This system says that the couple ( ; )p a

nullifies the action of the above operator D . We used in [48] an exterior calculus to establish a representation of the solution of system (5). Usually, the equations (5) are investigated as the Moisil-Teodorescu systems.

§ 3. Spatial integral representations For quaternionic-differentiable functions in the space three-dimensional space, we display an integral representation, which is a spatial counterpart of the Cauchy-Pompeiu formula (1). This representation was originated by G.C. Moisil [33].

Suppose that S is a two-dimensional closed Lyapunov surface 2) in ,3 denote by +Ω

the interior domain of S and by −Ω the exterior domain of S . Let ( , , )1 2 3n n n n=

be the

unit exterior normal at S and . ,2 3 1 1 3 2 1 2 3d n dS dx dx e dx dx e dx dx eσ = = + + where dS is the

surface element of .S The Gauss- Ostrogradski formula for one quaternion ( )pΦ gives

(6) .3

jjj 1

de dv

dx=

Φ = ∑ .

S

dσ Φ∫∫ ,

+Ω

∫∫∫ .3

jjj 1

de dv

dx=

Φ = ∑ .

S

dσΦ∫∫

and for two quaternions ( )pρ and ( )pΦ gives

+Ω

∫∫∫ ( . . ) dv =3

j

j 1

eρ=

Φ∑

S

∫∫ . .dρ σ Φ .

Taking into account of the differential rule of product

2 ) The conditions determining a Lyapunov surface are described in N.I. Muskhelishvili [39].

DAN PASCALI 108

( . . ) . . . .3 3 3

j j jj j jj 1 j 1 j 1

e e ex x x

ρρ ρ

= = =

∂ ∂ ∂Φ Φ = Φ + ∂ ∂ ∂

∑ ∑ ∑

we finally derive

(7)

+Ω

∫∫∫ . .3

jjj 1

e dvx

ρ

=

∂ Φ ∂ ∑ +

+Ω

∫∫∫ . .3

jjj 1

e dvx

ρ=

∂Φ = ∂ ∑ . .

S

dρ σ Φ∫∫ .

This formula will be used later for a representation of monogenic quaternions. Replacing the form of operators D and D in the relations (6) and (7), the Gauss- Ostro-gradski formula leads in the quaternionic case to the following equalities

(8)

S

d D dvσ

+Ω

⋅Φ = Φ∫∫ ∫∫∫ ,

S

d D dvσ

+Ω

Φ ⋅ = Φ∫∫ ∫∫∫

(9)

S

d D dv Dρ σ ρ ρ

+ +Ω Ω

⋅ ⋅Φ = ⋅ Φ + ⋅Φ∫∫ ∫∫∫ ∫∫∫ .

If ( )qΦ is a monogenic quaternion and ( )qρ is co-monogenic, we derive that

(10) ( ) ( ) ,q

S

q d q oρ σ⋅ ⋅Φ =∫∫

Let ( )P p be a fixed point in Ω and ( )Q q be a current point of the surface .S We denote

r PQ q p= = −

and calculate

( , )3

1 1 r 1P Q D D

r r r rrρ

= − = − = − =

.

For P Q≠ , we have

(11) 0.31

D Dr

ρ ρ

= = ∆ =

Let ( )P p be a point belonging to the interior domain ,+Ω bounded by the surface .S

We enclose the point P in a sphere ,Σ interior of ,+Ω and denote by ω the domain bounded

by .Σ Apply formula (8) to the domain ω+Ω and obtain

( ) ( ) ( ) .q q q

S

d q d q D q dv

ω ω

ρ σ ρ σ ρ

+Ω

⋅ ⋅Φ − ⋅ ⋅Φ = ⋅ Φ∫∫ ∫∫ ∫∫∫

Because ,r

d dSr

σ = we get

lim ( ) lim ( ) 4 ( )q q3 3r o r o

1 r 1q dS q dS p

rr rπ

→ →Σ Σ

− ⋅ ⋅Φ = Φ = Φ∫∫ ∫∫

and therefore for ( )P p +∈Ω we have


109

( )( ) ( )

4 4q

q

S

d1 1 1 1 D qp q dv

q p q p q p q p

σ

π π+Ω

ΦΦ = ⋅ ⋅Φ − ⋅

− − − −∫∫ ∫∫∫ .

For ( ) ,P p−∈Ω according to the equalities (8) and (10), we derive

( )

0 ( )4 4

qq

S

d1 1 1 1 D qq dv

q p q p q p q p

σ

π π+Ω

Φ= ⋅ ⋅Φ − ⋅

− − − −∫∫ ∫∫∫ .

In a concise form the above representations become ( ) ,

(12) ( , ) ( ) ( , ) ( )4 4 0 \ .

Q Q 3

S

P for P1 1P Q d Q P Q D Q dv

for Pρ σ ρ

π π+Ω

Φ ∈Ω⋅ ⋅Φ − ⋅ Φ =

∈ Ω∫∫ ∫∫∫

For ( , ),1C +Ψ ∈ Ω H the volume integral

( ) ( , ) ( )4 Q

1P p Q Q dvρ

π +ΩΠΨ = − ⋅ Ψ∫∫∫ ,

called the Moisil-Teodorescu operator, is continuous in ,3 a monogenic quaternion in

,−Ω vanishes at ,∞ and verifies the equation ( ) ( )D P PΠΨ = Ψ in .+Ω It plays a similar role of the Pompeiu operator (2) in the plane.

The formulae (12) can be used in a convenient way to obtain representations of solutions for various partial differential equations. Integral presentations for spatial models of mathematical physics are investigated by V.V. Kravchenko and M. V. Shapiro [25].

Finally, like in the plane, for any a regular solution of the iterated Dirac equation

(13) ( ) 0n 1D f P+ = in Ω is called the n-th order quaternionic areolar polynomial.

We will gives a representation of the .f Denoting ( , ) ( , )4

1K P Q P Qρ

π= − , we can write

( ) ( , ) ( ) QP K P Q Q dvΩ

ΠΨ = ⋅Ψ∫∫∫ and ( ) ( )D P PΠΨ = Ψ .

Let us consider the iterated operators ( , ) ( , ) ( , )n n 1RK P Q K P R K R P dv−

Ω= ⋅∫∫∫ and

( )n PΠ Ψ = ( , ) ( )nQK P Q Q dv

Ω⋅ Ψ∫∫∫ for , ,..n 2 3=

Since

( )n PΠ Ψ = ( )( , ) ( , ) ( ) ( ) ,n n 1 n 1R QK P R dv K R Q Q dv P− −

Ω Ω⋅ Ψ = Π Π Ψ∫∫∫ ∫∫∫

it follows that ,. . . , ,n n 1 n nD D−Π Ψ = Π Ψ Π Ψ = Ψ and, if ( )PΨ is monogenic, then

0.n 1D

+ Ψ = The above relations give for the solution of equation (13) a representation of the form

DAN PASCALI 110

(14) ( ) ( ) ( ) ( ) . . . ( ),2 no 1 2 nf P P P P P= Φ + ∏ Φ + ∏ Φ + + ∏ Φ

where ( ), , ,..., ,k P k o 1 nΦ = are monogenic quaternions in .Ω Moreover, taking into account

that

( )( ) ( ) ( ) ( ) ... ( ) ,k n k 1k k 1 k 2 nD f P P P P P− −

+ += Φ + ∏ Φ + ∏ Φ + + ∏ Φ

the monogenic quaternions, , ,..., ,k o 1 n= are determined by the values of ( )kD f Q on the boundary S , namely

( ) ( , ) ( ),4

kk Q

S

1P P Q d D f Qρ σ

πΦ = ⋅ ⋅∫∫ , ,..., .k o 1 n=

The representation (14) was establised by the author [45] and it remains valid in the n-dimensional case. The form (13) is compared with those involving the matrix algorithms due to M. Coroi-Nedelcu [14]. Analogue forms with the formula (14) in n-dimensional spaces are surveyed in [46] - [47]. By constructing suitable kernels, more detailed higher-order Cauchy - Pompeiu representations for maps with values in a universal Clifford algebras are obtained by H. Begher and his follower Z. Zhongxlang [66].

Variants of representation formula (11) and evaluations of three-dimensional integrals are discussed in [15]. In a version due to A. Perotti [49] for the study of quaternionic Cauchy-

Riemann in 3 attached a system of directional Hilbert operators on the sphere and the 3-

space H , while D.A. Pinotsis [50] extended the quaternionic formalism to more general classes of elliptic problems.

§4. Double integrals of Cauchy type

As above in §3, let S be a given closed Lyapunov surface and let us denote by be +Ω the

interior domain and by −Ω the exterior domain of .S A quaternionic-valued function :f S H is Hölder continuous on ,S if there are constants , ( , ]L o o 1α> ∈ such that

( ) ( )1 2 1 2f q f q L q qα

− ≤ − ( ), ( ) .11 2 2Q q Q q S∀ ∈

In the previous section, we have established that the Cauchy type integral

( ) ( , ) ( )4 Q

S

1p P Q d f Qρ σ

πΦ = ⋅ ⋅∫∫ = ( )

4q

S

d1 1f q

q p q p

σ

π⋅ ⋅

− −∫∫

defines a monogenic quaternion both in +Ω and in .−Ω When ( ) ( ) ,o oP p P p S→ ∈ this integral fails in the usual sense, and its main value (in

the Cauchy sense) will be determined. Indeed, let ε∑ be a sphere around oP S∈ of radius ε , denote by Sε the part of S

exterior of the sphere ε∑ and consider

( ) lim ( , ) ( )4o o Q

oS

1P P Q d f Q

ε

ερ σ

π→Φ = ⋅ ⋅ =∫∫ lim ( )

4q

o o oS

d1 1f q

q p q pε

ε

σ

π→⋅ ⋅

− −∫∫ .


111

Denoting by ε−∑ the part of the sphere ∑ exterior to the domain ,Ω we get

[ ]( ) lim ( , ) ( ( ) ( )) ( ) ( , ) ( )4 4o o q o o o q o

oS

1 1P P Q d f Q f P f P P Q d f P

ε ε

ερ σ ρ σ

π π−

→

∑

Φ = ⋅ ⋅ − + − ⋅ ⋅∫∫ ∫∫ .

As ( )f Q is a Hölder continuous function, the above limit exists and

( )( ) ( , ) ( ) ( ) ( , ) ( ) ( ) .4o o Q o o q o

S S

1 1 1P P Q d f Q f P P Q d f Q f P

2 2ρ σ ρ σ

πΦ = ⋅ ⋅ = + ⋅ ⋅ −∫∫ ∫∫

Similarly in the plane case, one can establish formulas for the jump on boundary for the integral of Cauchy type. The Cauchy integral can be written as

( )( ) ( , ) ( ) ( ) ( , ) ( )4 4Q o Q o

S S

1 1p P Q d f Q f P P Q d f Pρ σ ρ σ

π πΦ = ⋅ ⋅ − + ⋅ ⋅∫∫ ∫∫

whence

( )( ) ( , ) ( ) ( ) ( )4 Q o o

S

1p P Q d f Q f P f Pρ σ

πΦ = ⋅ ⋅ − +∫∫ for ,P

+∈Ω

and

( )( ) ( , ) ( ) ( )4 Q o

S

1p P Q d f Q f Pρ σ

πΦ = ⋅ ⋅ −∫∫ for .P

−∈Ω

Therefore

(15) ( ) lim ( ) ( ) ( , ) ( ),4o

o o o QP P

SP

1 1P P f P P Q d f Q

2ρ σ

π+

+

→

∈Ω

Φ = Φ = + ⋅ ⋅∫∫

(16) ( ) lim ( ) ( ) ( , ) ( ).4o

o o o QP P

SP

1 1P P f P P Q d f Q

2ρ σ

π−

−

→

∈Ω

Φ = Φ = − + ⋅ ⋅∫∫

Finally, from (15) and (16), we deduce the Plemelj - Sokhotzki formulas

(17) ( ) ( ) ( ),o o oP P f P+ −Φ − Φ =

(18) ( ) ( ) ( , ) ( ).o o o Q

S

1P P P Q d f Q

2ρ σ

π+ −Φ + Φ = ⋅ ⋅∫∫

Theorem 1. . In order that a quaternionic-valued function f and Hölder continuous on

S , to represents the boundary-value of a monogenic quaternion in +Ω it is necessary and

sufficient that one of the following conditions

(19) ( , ) ( )4 Q

S

1P Q d f Q oρ σ

π⋅ ⋅ =∫∫ for all ,P

−∈Ω

or

DAN PASCALI 112

(20) ( ) ( , ) ( )o o q

S

1 1f P P Q d f Q o

2 2ρ σ− + ⋅ ⋅ =∫∫ for all ,oP S∈

be satisfied. Necessity. Let Φ be a monogenic quaternion, verifying the Hölder continuity on S and

taking the value ( )f Q on .S Then

( ) ,( , ) ( )

4 0 ,Q

S

P for P1P Q d f Q

for Pρ σ

π

+

−

Φ ∈Ω⋅ ⋅ =

∈Ω∫∫

whence the necessity of the conditions follows.

Sufficiency. In virtue of equations (19)-(20) and relation (16), it follows that ( ) 0,oP−Φ =

and taking into account of (17), we deduce ( ) ( ) ( ) ( ).o o o of P P P P+ − += Φ − Φ = Φ

In a similar way, the following statement can be shown Theorem 2. In order that a quaternionic-valued function f, Hölder continuous on S to

represent the boundary-value of a monogenic quaternion in −Ω it is necessary and

sufficient that one of the following conditions

( , ) ( )4 Q

S

1P Q d f Q oρ σ

π⋅ ⋅ =∫∫ for all ,P

+∈Ω

or

( ) ( , ) ( )o o q

S

1 1f P P Q d f Q o

2 2ρ σ+ ⋅ ⋅ =∫∫ for all ,oP S∈

be statisfied.

In conclusion, we establish The Poincaré-Bertrand formula. If the quaternion ( )QΦ is defined and verifies a

Hölder condition on ,S then the superposition of singular integrals

(21) ( , ) ( , ) . ( ) ( )4 1o Q 1 Q 1 o2

S S

1P Q d Q Q d Q Pρ σ ρ σ

π⋅ ⋅ ⋅ Φ = Φ∫∫ ∫∫ .oP S∀ ∈

holds.

Proof. Denote

(22) ( ) ( , ) ( )41 o o Q

S

1P P Q d Qρ σ

πΦ = ⋅ ⋅Φ∫∫

(23) ( ) ( , ) ( )42 o o Q 1

S

1P P Q d Qρ σ

πΦ = ⋅ ⋅Φ∫∫

and check easily that ,1 2Φ Φ are Hölder continuous functions on .S Consider now the of

Cauchy type integrals


113

(24) ( ) ( , ) ( )4 Q

S

1P P Q d Qρ σ

πΨ = ⋅ ⋅Φ∫∫ and ( ) ( , ) ( )

41 Q 1

S

1P P Q d Qρ σ

πΨ = ⋅ ⋅Φ∫∫ .

By virtue of (16), we infer that

( ) ( ) ( , ) ( ),4o o o Q

S

1 1P P P Q d Q

2ρ σ

π+Ψ = Φ + ⋅ ⋅Φ∫∫

( ) ( ) ( , ) ( ).41 o 1 o o Q 1

S

1 1P P P Q d Q

2ρ σ

π+Ψ = Φ + ⋅ ⋅Φ∫∫

According to equations (22)-(23), it follows that

(25) ( ) ( ) ( )1 o o o1

P P P2

+Φ = Ψ − Φ and ( ) ( ) ( ).2 o 1 o 1 o1

P P P2

+Φ = Ψ − Φ

Replacing 1( )QΦ in (24), we obtain

( ) ( , ) ( ) ( )41 Q

S

1 1P P Q d Q Q

2ρ σ

π+

Ψ = ⋅ ⋅ Ψ − Φ = ∫∫ ( , ) ( )4 Q

S

1P Q d Qρ σ

π+⋅ ⋅ Ψ −∫∫

( , ) ( )4 Q

S

1 1P Q d Q

2ρ σ

π− ⋅ ⋅ ⋅Φ =∫∫ ( ) ( ) ( )

1 1P P P

2 2Ψ − Ψ = Ψ

and, as a result, ( ) ( ).1 o o1

P P2

+ +Ψ = Ψ Finally, by means of (25), we deduce that

( ) ( ) ( ) ( ) ( )2 o o o o o1 1 1 1

P P P P P2 2 2 2

+ + Φ = Ψ − Ψ − Φ = Φ

and the equality (23) implies the formula (21). Spatial analogue the principal value of a singular three-dimensional integral of Cauchy

type considered first time, in the matrix framework by A.V.Bicadze [10].The basic techniques

of the Plemelj-Sokhotskiĭ type and the Poincaré-Bertrand formula are extended to 3 .These

results are presented later by R. Abreu-J. Reyes [4], by E. Obolashvili [40] with Clifford analytic methods and generalized to piecewise Lyapunov surfaces by B.Schneider [56].

§ 5. Inversion of the Bicadze system of singular integral equations

Let S be a closed Lyapunov surface in 3 and ( , )P Qρ the kernel defined in the previous

sections. We look now to the solutions, Hölder continuous on ,S of the general singular equation

(26) ( , ) ( ) ( )o Q o

S

1P Q d Q P

2ρ σ

π⋅ ⋅Φ = Ψ∫∫ ,oP S∀ ∈

where ( )QΨ is a given quaternionic function and S is a closed Lyapunov surface. By virtue of the Poincaré-Bertrand formula of superposition of singular integrals (21), a solution of equation (26) is given by

DAN PASCALI 114

(27) ( ) ( , ) ( )o o Q

S

1P P Q d Q

2ρ σ

πΦ = ⋅ ⋅Ψ∫∫ .oP S∀ ∈

Reciprocally, a solution of singular integral equation (27) has the form (26) and hence (26)-(27) represent the inversion formulas. Later on, we explain a method, due to Bicadze [11], for the solution of the singular integral equation

(28) b

a ( ) ( , ) ( ) ( )o o Q

S

P P Q d Q f Q2

ρ σπ

⋅Φ + ⋅ ⋅ ⋅Φ =∫∫ ,oP S∀ ∈

where a, b are given constant quaternions, f is a known quaternion, Hölder continuous on S. We suppose only that a b≠ ± and define the piecewise monogenic quaternion

( ) ( , ) ( ).Q

S

1P P Q d Q

2ρ σ

πΨ = ⋅ ⋅Φ∫∫

By virtue of the Plemelj-Sokhotzki formulas, one can write for oP S∈ that

( ) ( ) = ( )o o oP P P+ −Ψ − Ψ Φ

(29)

( ) ( ) ( , ) ( )o o o Q

S

1P P P Q d Q

2ρ σ

π+ −Ψ + Ψ = ⋅ ⋅Φ∫∫ ,

while the equation (28) becomes

(a+b) ( ) (a-b) ( ) ( )o o oP P f P+ −⋅Ψ = ⋅ Ψ +

or

(30) ( ) G ( ) g( )o o oP P P+ −Ψ = ⋅Ψ +

where

G = (a+b) (a-b)1− ⋅ and g( ) (a+b) ( ).1o oP f P

−= ⋅

This problem is similar to the Hilbert problem in the monogenic function theory of one complex variable.

Besides, the quaternionic function

,( )o

G for PP

1 for P

+

−

∈ΩΨ =

∈Ω

is a solution of a homogeneous problem ( ) G ( )o oP P+ −Ψ = ⋅Ψ

( ) ( )or ( ) ( )1 1

1o o o oP P G

− −+ + −

Ψ = Ψ ⋅

and the inhomogeneous problem (30) can be written in the form

( ) ( ) ( ).( ) ( ) ( )

o o o

o o o o o o

1 1 1P P g P

P P P

+ −+ − +

⋅ Ψ − ⋅Ψ = ⋅Ψ Ψ Ψ

The quaternion


115

( ) ( )( )o

1F P P

P= ⋅ Ψ

Ψ

is piecewise monogenic and

( ) ( ) ( ).( )o o o

o o

1F P F P g P

P

+ −− = ⋅Ψ

Consequently,

( ) ( , ) ( )4 ( )

Q

oS

1 1F P P Q d g Q

Qρ σ

π += ⋅ ⋅ ⋅

Ψ∫∫

and ( )

( ) ( , ) ( ),4 ( )o

Q

oS

P 1P P Q d g Q

Qρ σ

π +

ΨΨ = ⋅ ⋅ ⋅

Ψ∫∫

whose boundary-values (15)-(16) are

( ) (a b) ( ) (a+b) ( , ) ( ),4

1 1o o o Q

S

1 1P f P P Q d f Q

2ρ σ

π+ − −Ψ = + ⋅ + ⋅ ⋅∫∫

( ) (a-b) ( ) (a-b) ( , ) ( ).4

1 1o o o Q

S

1 1P f P P Q d f Q

2ρ σ

π− − −Ψ = − ⋅ + ⋅ ⋅∫∫

By virtue of (29), a general form of the solution of equation (28) is given by

( ) (a+b) + (a-b) ( ) (a+b) (a-b) ( , ) ( ).4

1 1 1 1Q

S

1 1P f P P Q d f Q

2ρ σ

π− − − − Ψ = ⋅ + − ⋅ ⋅

∫∫

For the solution of equation (28), by means of order 4 matrix calculus, where a and b are order 4 constant matrices, A.V. Bicadze [11] has required the following conditions:

a) det (a+b) o≠ and det (a-b) o≠ ;

b) The matrix (a+b) (a-b)1G

−= has the form

4

4

4

4

.

1 2 3

2 1 3

3 1 2

3 2 1

g g g g

g g g gG

g g g g

g g g g

− −=

− −

− −

The replacement of the above condition by a single one shows the advantage of using the quaternionic algorithm. This assumption is related to the matrix representation of quaternions. S. Bernstein [7]-[9] and R. Abreu-Blaya - J. Bory-Reyes [3]-[5] adapted the research on Cauchy spatial integrals and quaternionic Riemann-Hilbert problems to the framework of Clifford analysis and the current singular integral operator theory. The cases in domains with a piece-wise Lyapunov surface are investigated in [6].

DAN PASCALI 116

§ 6. Comments Later on, we insert some distinctive features about great Romanian scholars interested in the areolar derivative and its extensions to the higher-dimensional spaces.

On March 31, 1905, Dimitrie Pompeiu defended at Sorbonne his P.D.-thesis about the continuity of functions of complex variables. An essential result of his dissertation attested the existence of uniform analytic functions continuous on the set of its singularities. This assertion was initial disputed by Ludovic Zoretti, a disciple of Emilie Borel, who claimed that a uniform function is necessarily discontinuous on the set of singularities. Soon Zoretti’s affirmation was proved to be untrue and Arnaud Denjoy, one of the great French mathematician of that times published in Comptes Rendus (1909), a note showing that the arguments of Pompeiu are correct. Moreover, in his thesis, Pompeiu set out on preliminaries for the definition of set distance [1], developed later in the works of F. Hausdorff and C. Kuratowski.

On the subject of our research, Dimitrie Pompeiu [58] introduced the areolar derivative and established the crucial Cauchy-Pompeiu representation (1).

After my arrival in the USA, I took part, for almost two decades, at the seminar of non-

linear analysis, organized by A. Bahri and H. Brézis. at the Rutgers University, New Brunswick, New Jersey. As a native Romanian and educated in Romania, I felt proud several times when I heard that the treated subjects are roots in the scientific investigation of Gheorghe Tițeica and Dimitrie Pompeiu., because they were the first Romanian mathe-maticians with doctorates at Paris on the early of the 20-th century 3).

In 1950’s, L. Bers displayed the pseudo-analytic function theory, concurrently with the construction of the theory of generalized analytic functions by I.N. Vekua. In particular, the publication of I.N. Vekua’s book [63] revived the interest of Romanian mathematicians for the areolar derivative theory, both in the complex plane and in the higher dimensional spaces. Thus, the inhomogeneous Cauchy-Riemann systems for the complex function f u iv= + ,with the linear right part in u and v , is written as Vekua’s equation (31) ( ) ( )z f a z f b z f∂ = + .

and its solutions are called generalized analytic functions. In the case of one unknown function, I.N. Vekua established a similarity principle, which means that, the solution of the equation (31) differs from an analytic function by a smooth exponential factor, depending only on the coefficients a and .b This principle reduces many boundary-value problems attached to the equation (31) to the corresponding ones for analytic functions.

The first attempt, carried out by D. Pascali [44] in 1965, was to generalize the similarity principle to the system for generalized analytic vectors (32) ( ) ( ) ,zW A z W B z W∂ = +

with complex-valued n n× matrices A and ,B which means that any solution of (32) has the form W S ψ= ⋅ , where S is a nonsingular continuous matrix and ψ is an analytic vector. Later, due to this simple representation of solutions, W. Wendland [64] called “Pascali’s systems” the equations (b), whose solutions have the form W S ψ= ⋅ . Later, J.L. Buchanan [13] in 1982 and U. Kuzman [26] in 2012 improved in their Ph.D. theses the similarity

3 ) For example, M.S. Vogolis work [64] references contains four quotations involving Pompeiu’s problem.


117

principle w S ψ= ⋅ in general settings. The Pascali’s systems are considered in the book [20], and paper [2], as well.

In the Romanian mathematics of the 1960’s, regarding extensions of the areolar derivative theory to Euclidean n-dimensional spaces by using the hypercomplex-valued functions, we re-mark the comprehensive studies of V. Iftimie. The first paper [26] describes similar Cauchy-Pompeiu representations and a multidimensional analogue of the Cauchy integral singular operator with applications to the solution of the Poincaré-Stekloff problem and the inversion of H. Villat integral equation. Properties of Pompeiu operator (areolar primitive), extending (1) to higher-dimensional spaces are presented by V. Iftimie [27], conform with Mikhlin’s techniques [31] for multidimensional singular integral operators and in connection with the modern Calderȯn-Zigmund theory. The accurateness of computations, used by V.Iftimie in his extensions of the areolar derivative theory to multi-dimensional spaces, as well as a complete list of Romanian investigations in this field, in this field prior to 1965 must be underlined. In recent decades the field of hypercomplex analysis has been dominated by Clifford algebras. This topic provides a suitable and powerful framework to tackle problems of harmonic analysis by procedures similar to one complex variable, but now in higher-dimensional spaces. Thus, S. Gal [17] examined some geometric peculiarities of hyper-complex functions.

H. Malonek [28] narrated a delightful story of R. Fueter’s life, fascinated of the noncom-muntative multiplication rule for the quaternions of Sir William Rowan Hamilton [22]. As a beneficiary of a Alexander von Humboldt fellowship, Malonek’s account remembered me the spent periods and particularly the musical parties organized at the German or Swiss departments of mathematics. Certainly, Fueter’s contributions [16] have been quoted by D. Pascali [42] and V. Iftimie [23] in 1960’s.

Unlike the previous scholars, Gr.C. Moisil defended his PhD thesis (1929) in Romania under the supervision of Dimitrie Pompeiu and Gh. Țițeica. The family, adolescence, education of Gr. C. Moisil are described in the biographic book written his wife Viorica Moisil [38] and in the essay introducing his volumes “Mathematical Works” [35]-[36], published by S. Marcus. More-over, a comprehensive display on Moisil’s investigations in algebraic logic is reported by his followers G. Georgescu, A. Iorgulescu and S. Rudeanu [19].

Along his life Gr.C. Moisil had preoccupations related to the complex functions, theoretical mechanics, and partial differential equations, the majority of his contributions are included in the volumes [35]-[36]. Solomon Marcus have supplied an competent presentation [30] of the academic activity of Gr. C Moisil. Less mentioned, the quaternionic analysis initiated by Gr. C. Moisil [33] - [34] in 1931 is revealed in the first part of this work.

Acknowledgments

The house of Professor Dimitrie Pompeiu was situated in the semi-central residential district Tei in Bucharest at the intersection of the Barbu Văcărescu and Eugen Botez streets. His mother-in-law Mrs. Ionescu dwelled in nearby, (3 Buestrului St.) in a twin house with my uncle, the painter Rudolf Schweitzer-Cumpăna. Hearing that I became a researcher at the Institute of Mathematics, Petronia, the daughter of Mrs. Ionescu and the wife of D. Pompeiu, made me in 1959 a present, with a nice dedication, of a copy of the volume ”Mathematical

DAN PASCALI 118

Works” of Dimitrie Pompeiu. Indeed, Pompeiu’s papers on the areolar derivative played the pivot of my Ph.D-thesis „Representation of Solutions of Equations with Areolar Derivatives”, defended in 1964, under the guidance of M. Nicolescu and referents G.C. Moisil and N.V. Teodorescu.

After my traveling through European countries and several states in SUA, in my private library there are in an honour place the volumes of mathematical works of Dimitrie Pompeiu, Miron Nicolescu and Gr. C.Moisil, all edited by Acad. Solomon Marcus. Nicolae V.Teodo-rescu’s life and science activity is presented by E. Otlâcan [44], as well. REFERENCES

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Abstracts of some paperspresented to conference




A special class of LΦ,ϕ operators

Anca ARMASELU1

Abstract

By taking a special class of functions used by N. Tita in the paper“LΦ,ϕ operators and (Φ, ϕ) spaces”, (Collect Math) 30, p. 3-10, 1979,one obtains a class of operators with some specific properties that donot characterize all classes LΦ,ϕ.

1Department of Electrical Engineering and Applied Physics, Transilvania University ofBrasov, Romania, e-mail: anca [email protected]



On some multilinear and bounded operators

Nicolae TITA1

Abstract

This paper presents several new aspects of boundedness of classesof k-linear operators (k ≥ 2). Also are some remarks about a previouspaper ”On some limit scales of approximation ideals“, Proc. ICNAAM(2011) 536-539.

Precisely, in the inequality:

k∑

n=1

log

(1 + an

(r∏

i=1

Ti

))≤ r

k∑

n=1

r∏

i=1

log(1 + an(Ti)),

the factor r is in fact c(r), where

c(r) ≤ rr∏

i=1

(1 + ‖Ti‖);

(an(T )) is the sequence of the approximation numbers of T ∈ L(X)and X is a normed space.

This results from the fact that

k∑

n=1

log

(1 + an

(r∏

i=1

Ti

))≤

k∑

n=1

an

(r∏

i=1

Ti

)≤ r

k∑

n=1

r∏

i=1

an(Ti)

and the relation

an(Ti) ≤ (1 + ‖Ti‖) log(1 + an(Ti)), i = 1, 2, . . . , r.

The similar observation is valid for the remark 03 and proposition06.

1Department of Mathematics, Transilvania University of Brasov, Romania, e-mail:[email protected]

On some multilinear and bounded operators 124

For the case of tensor product operators are the inequalities:

k∑

n=1

log(1 + an(S ⊗ T ))

n≤ 6

k∑

n=1

‖S‖ log(1 + an(T )) + ‖T‖ log(1 + an(S))

n,

k = 1, 2, . . ., if ‖S‖, ‖T‖ ≥ 1If S and T are contractions we obtain:

k∑

n=1

log(1 + an(S ⊗ T ))

n≤ 6

k∑

n=1

log(1 + an(S))(1 + an(T ))

n.

2000 Mathematics Subject Classification: 47B06, 47L20.Key words: approximation numbers, operator ideals.

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