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Jordan Journal of Mathematics and Statistics An International Peer-Reviewed Research Journal

Volume 1, No. 1, April 2008, Rabia 2 1429 H

Jordan Journal of Mathematics and Statistics (JJMS): An International Peer-Reviewed Research Journal established by the Higher Research Committee, Ministry of Higher Education & Scientific Research, Jordan, and published quarterly by the Deanship of Research & Graduate Studies, Yarmouk University, Irbid, Jordan. EDITOR-IN-CHIEF: Mashhoor A. Al-Refai

Department of Mathematics, Yarmouk University, Irbid, Jordan. Current Address:Vice President, Yarmouk University, Irbid, Jordan.

E-mail: [email protected] EDITORIAL SECRETARY: Mrs. Manar M. Malkawi, Deanship of Research and Graduate Studies EDITORIAL BOARD:

Nabil T. Shawagfeh Department of Mathematics, University of Jordan, Amman, Jordan.

Current Address: President, Al-al-Bayt University, Al-Mafraq, Jordan. E-mail: [email protected] Mufid M. Azzam Department of Mathematics, University of Jordan, Amman, Jordan. E-mail: [email protected] Ahmed T. Alawneh Department of Mathematics, University of Jordan, Amman, Jordan. E-mail: [email protected] Bassam Y. Al-Nashef Department of Mathematics, Yarmouk University, Irbid, Jordan. E-mail: [email protected] Fuad A. Kittaneh Department of Mathematics, University of Jordan, Amman, Jordan. E-mail: [email protected] Abdullah M. Al-Jarrah Department of Mathematics, Yarmouk University, Irbid, Jordan. E-mail: [email protected]

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Jordan Journal of

Mathematics and Statistics An International Peer-Reviewed Research Journal


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Jordan Journal of Mathematics and Statistics (JJMS) is an international quarterly peer-reviewed research journal issued by the Higher Scientific Research Committee, Ministry of Higher Education and Scientific Research, Amman, Jordan. JJMS is published by the Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan. The Journal endeavors to publish significant original research articles in all areas of Pure Mathematics, Applied Mathematics, Pure Statistics , Applied Statistics and other related areas. Survey Papers and short communications will also be considered for publication.

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The Hashemite Kingdom of Jordan Yarmouk University





TABLE of CONTENTS

Modern Functional Analysis in Theory of Sequence Spaces and Matrix Transformations Eberhard Malkowsky

1

Strongly Singular Calderon-Zygmund Operators and their Commutators Yan Lin and Shanzhen Lu

31

Matrix Transformations between Sets of the form Wξ and Operator Generators of Analytic Semigroups Bruno De Malafosse and Eberhard Malkowsky

51

Best Simultaneous Approximation in Metric Spaces Sharifa Al-Sharif

69

Algebraic Models in Applied Research Thomas Vougiouklis and Penelope Kambaki

81

The Set of Values of Functional in the Classes of Functions having Integral Representation Hassan Baddour

91

Jordan Journal of Mathematics and Statistics (JJMS) 2008, 1(1), pp.1-29

MODERN FUNCTIONAL ANALYSIS IN THE THEORY OFSEQUENCE SPACES AND MATRIX TRANSFORMATIONS

EBERHARD MALKOWSKY

Abstract. Many concepts and theories in functional analysis have turned out to bepowerful and widely used tools in operator theory, in particular in the theory of matrixtransformations between sequences spaces in summability. We give an introduction tothe basic theory of FK, BK, AK and AD spaces, the various types of dual spaces ofsequence spaces, and apply the general results to the characterisations of classes of ma-trix operators between certain sequence spaces that arise in summability. We also studythe Hausdorff measure of noncompactness and its applications to the characterisationsof compact operators between sequence spaces.

This is a survey paper which also includes some results of the author’s joint researchwith V. Rakocevic and I. Djolevic at the Department of Mathematics of the Facultyof Science and Mathematics at the University of Nis, Serbia. Although many of theresults are probably known to specialists, the proofs are included for the convenience ofthose readers who may not be too familiar with the subject, and an appendix is addedat the end containing the fundamental theorems in functional analysis in the versionsthey are applied.

1. Introduction, Standard Notations and Well–Known Results

Let X be a normed space. Then we denote the open unit ball and the unit sphere inX by BX = {x ∈ X : ‖x‖ < 1} and SX = {x ∈ X : ‖x‖ = 1}.

Let X and Y be Banach spaces. Then B(X, Y ) denotes the set of all bounded linearoperators L : X → Y ; B(X, Y ) is a Banach space with the operator norm defined by‖L‖ = supx∈SX

‖L(x)‖ for all L ∈ B(X, Y ).Let X be a linear metric space. A Schauder basis of X is a sequence (bn)∞n=0 in

X such that, for every x ∈ X, there exists a unique sequence (λn)∞n=0 of scalars suchthat x =

∑∞n=0 λnbn. By X ′ we denote the continuous dual of X, that is the set of all

continuous linear functionals on X. If X is a Banach space then we write X∗ for X ′

with its norm defined by ‖f‖ = supx∈SX|f(x)|.

We write ω, c0, c and `∞ for the sets of all complex, null, convergent and boundedsequences, `p = {x ∈ ω :

∑∞k=0|xk|p < ∞} for 1 ≤ p < ∞, and cs and bs for the sets of

1991 Mathematics Subject Classification. Primary: 40H05; Secondary: 46A15.Key words and phrases. Summability, sequence spaces, FK, BK, AD, AK spaces, dual spaces,

matrix transformations, measures of noncompactness.Work supported by the research grants #1232 and #144003 of the Serbian Ministry of Science,

Technology and Development.Copyright c© Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan.Received Sept. 1, 2007, Accepted March 4, 2008.

1

2 EBERHARD MALKOWSKY

all convergent and bounded series. By e and e(n) (n = 0, 1, . . . ), we denote the sequences

with ek = 1 for all k, and e(n)n = 1 and e

(n)k = 0 for k 6= n.

It is well known that ω is a complete locally convex linear metric space with its metricgiven by

(1.1) d(x, y) =∞∑

k=0

1

2k

|xk − yk|1 + |xk − yk| for all x = (xk)

∞k=0, y = (yk)

∞k=0 ∈ ω;

c0, c, `∞, `p (1 ≤ p < ∞), cs and bs are Banach spaces with their natural normsgiven by ‖x‖∞ = supk |xk| on c0, c and `∞, ‖x‖p = (

∑∞k=0|xk|p)1/p on `p, and ‖x‖bs =

supn |∑n

k=0 xk| on cs and bs.Furthermore, c∗0 is norm isomorphic to `1; this means f ∈ c∗0 if and only if f(x) =∑∞k=0akxk (x ∈ X) for some a ∈ `1, and ‖f‖ = ‖a‖ ([Wil1, Example 6.4.4, p. 91]).

Similarly `∗1 is norm isomorphic to `∞ ([Wil1, Example 6.4.2, p. 91]), and `∗p for 1 < p <∞ is norm isomorphic to `q where q = p/(p − 1) ([Wil1, Example 6.4.3, p. 91]); alsof ∈ c∗ if and only if

(1.2) f(x) = χf limk→∞

xk +∞∑

k=0

akxk (x ∈ c) with a = (f(e(k))∞k=0 ∈ `1,

where

(1.3) χf = f(e)−∞∑

k=0

f(e(k)) ([Wil1, Example 6.4.5, p. 92]);

furthermore

(1.4) ‖f‖ = |χf |+ ‖a‖1.

Finally, the continuous dual of `∞ is not given by a sequence space ([Wil1, Example6.4.8, pp. 93, 94]).

We give a short survey of the most important concepts and methods of summability,an introduction to the basic theories of FK, BK, AK and AD spaces, consider multi-plier and dual spaces of sequence spaces, characterise matrix transformations betweensequence spaces, and apply the Hausdorff measure of noncompactness to the character-isations of compact operators between sequence spaces.

2. Summability

This section is intended as a motivation of what follows; the results presented here arenot needed in the sequel.

Summability encompasses a variety of fields, originally mainly from analysis, and hasmany applications, for instance in numerical analysis to speed up the rate of convergence,and in approximation theory, operator theory and the theory of orthogonal series.

SEQUENCE SPACES AND MATRIX TRANSFORMATIONS 3

2.1. Concepts and Methods of Summability. The classical summability theory dealswith a generalisation of the convergence of sequences or series of real or complex numbers.The idea is to assign a limit to divergent sequences or series by considering a transformrather than the original sequence or series. Most popular are matrix transformationsgiven by an infinite matrix A = (ank)

∞n,k=0.

There are three concepts of summability, ordinary, absolute and strong summability.First we consider ordinary summability. A sequence x = (xk)

∞k=0 of complex numbers

is said to be summable A to a complex number η, if the series

(2.1) Anx =∞∑

k=0

ankxk converge for all n and limn→∞

Anx = η;

this is denoted by x → η(A). The matrix A defines a summability method A or a matrixtransformation by (2.1).

Let 0 < p < ∞. Then a sequence x is said to be absolutely summable with index p toa complex number η if the series Anx in (2.1) converge for all n, and

∑∞n=0 |Anx|p = η;

this is denoted by x → η|A|p. A sequence x is said to be strongly summable A withindex p to a complex number ξ if the series

∑∞k=0ank|xk − ξ|p converge for all n and

limn→∞∑∞

k=0ank|xk − ξ|p = 0; this is denoted by x → ξ[A]p.

Example 2.1. Let the matrix A = (an,k)∞n,k=0 be given by ank = 1/(n + 1) for 0 ≤ k ≤ n

and ank = 0 (n = 0, 1, . . . ). Then A transforms every sequence x into the sequence of itsarithmetic means. It is well known by Cauchy’s theorem that every convergent sequenceis summable A to the same limit. Furthermore, the divergent sequence ((−1)k)∞k=0 issummable A to 0.

The most important summability methods are given by Hausdorff matrices and theirspecial cases, the Cesaro, Holder and Euler matrices, and by Norlund matrices. Allthese matrices are triangles, that is ank = 0 for k > n and ann 6= 0 (n = 0, 1, . . . ).

Let µ = (µ)∞n=0 be a given complex sequence, M = (mnk)∞n,k=0 be the diagonal matrix

with mnn = µn (n = 0, 1, . . . ), and D = (dnk)∞n,k=0 be the matrix with dnk = (−1)k

(nk

).

Then the matrix H = H(µ) = DMD is called the Hausdorff matrix associated withthe sequence µ; its entries are given by hnk =

∑nj=k(−1)j+k

(nj

)(jk

)µj (0 ≤ k ≤ n; n =

0, 1, . . . ). The Cesaro matrix Cα of order α > −1 is the Hausdorff matrix associatedwith the sequence µ where µn = Aα

n =(

n+αn

)(n = 0, 1, . . . ); its entries are given by

(Cα)n,k = Aα−1n−k/A

αn (0 ≤ k ≤ n; n = 0, 1, . . . ); the numbers Aα

n are called the Cesarocoefficients of order α. The Holder matrix Hα of order α > −1 is the Hausdorff matrixassociated with the sequence µ where µn = (n + 1)−α (n = 0, 1, . . . ); no explicit formulais known for the entries of the matrices Hα, in general. The Euler matrix Eq of orderq > 0 is the Hausdorff matrix associated with the sequence µ where µn = (q +1)−n (n =0, 1, . . . ); its entries are given by (Eq)n,k =

(nk

)qn−k(q + 1)−n (0 ≤ k ≤ n; n = 0, 1, . . . ).

Finally, let q = (qk)∞k=0 be a sequence of complex numbers such that Qn =

∑nk=0 qk 6= 0

for all n. Then the Norlund matrix (N, q) is given by ((N, q))n,k = qn−k/Qn (0 ≤ k ≤n; n = 0, 1, . . . ).


Example 2.2. (a) Let µ = e. Then we obtain for the Hausdorff matrix H = H(µ)

hnk =n∑

j=k

(−1)j+k

(n

j

)(j

k

)µj =

(n

k

) n∑

j=k

(−1)j+k

(n− k

j − k

)

=

(n

k

) n−k∑j=0

(−1)k

(n− k

j

)= δnk (n, k = 0, 1, )

where δnn = 1 and δnk = 0 for k 6= 0. Thus we have H = I, the identity matrix.(b) Let µn = 1/(n + 1) = A1

n (n = 0, 1, . . . ).Then we obtain H(u) = H1 = C1; thus thematrix A of Example 2.1 is a Holder and Cesaro matrix of order 1.(c) Let q = e. Then we obtain Qn = n+1 (n = 0, 1, . . . ). Thus the matrix A of Example2.1 is also the Norlund matrix (N, e).

We refer the interested reader to [Boo, Coo, Har, Mad, Pey, Z–B] for the classicalsummability theory.

2.2. Matrix Transformations. The theory of matrix transformations deals with es-tablishing necessary and sufficient conditions on the entries of a matrix to map a sequencespace X into a sequence space Y . This is a natural generalisation of the problem to char-acterise all summability methods given by infinite matrices that preserve convergence.

Given X, Y ⊂ ω, we write (X, Y ) for the class of all infinite matrices that map X intoY . So A ∈ (X, Y ) if and only if the series Anx in (2.1) converge for all n and all x ∈ X,and

(2.2) Ax = (Anx)∞n=0 ∈ Y for all x ∈ X.

The first results were the Toeplitz theorem for the classes (c, c) of conservative or (con-vergence preserving) matrices, (c, c; P ) of regular matrices, that is conservative matricesthat preserve limits, and the Schur theorem for the classes (`∞, c), the so–called coercivematrices, and (`∞, c0).

Theorem 2.3 (O. Toeplitz, 1911). ([Toe]) (a) We have A ∈ (c, c) if and only if

(i) ‖A‖ = supn

∞∑

k=0

|ank| < ∞, (ii) limn→∞

ank = αk exists for every k and (iii) limn→∞

∞∑

k=0

ank = α exists.

If A ∈ (c, c) and x ∈ c then

limn→∞

An(x) =

(α−

∞∑

k=0

αk

)limk→∞

xk +∞∑

k=0

αkxk.

(b) We have A ∈ (c, c; P ) if and only if (i), (ii) and (iii) in (a) hold with αk = 0(k = 0, 1, . . . ) and α = 1.

Theorem 2.4 (O. Schur, 1920). ([Wil2, Theorem 1.7.18, p. 15]) (a) We have A ∈ (`∞, c)if and only if (ii) in Theorem 2.3 holds and

(i’) supn

∞∑

k=0

|ank| is uniformly convergent in n.


(b) ([S–T, 21, (21.1)]) We have A ∈ (`∞, c0) if and only if

(ii’) limn→∞

∞∑

k=0

|ank| = 0.

We close this section with some applications of Theorems 2.3 and 2.4.

Example 2.5. (a) The matrix A of Example 2.1 is regular.(b) The Euler matrices Eq are regular for all q > 0.

Proof. (a) This is obvious from Theorem 2.3 (b).(b) We write A = Eq. Since ank ≥ 0 (n, k = 0, 1, . . . ) for q > 0, it follows that

∞∑

k=0

|ank| =∞∑

k=0

ank =1

(q + 1)n

n∑

k=0

(n

k

)qn−k =

(q + 1)n

(q + 1)n= 1 for all n,

and (i) and (iii) of Theorem 2.3 (b) are satisfied. We fix k. Since 0 < q/(q + 1) < 1, wehave, for ρ = 1/q > 0, q/(q + 1) = 1/(1 + ρ), and so

0 ≤ ank =1

(q + 1)n

(n

k

)qn−k =

1

qk

(n

k

)1

(1 + ρ)n≤ 1

qk

(n

k

)1(

nk+1

)ρk+1

=1

ρ(qρ)k

k + 1

n− k→ 0 (n →∞).

Thus (ii) of Theorem 2.3 (b) is also satisfed. ¤

Example 2.6. The famous Steinhaus theorem states that, for every regular matrix A,there is a bounded sequence which is not summable A.

Proof. We assume there is a matrix A ∈ (c, c; P )∩(`∞, c). Then it follows from Theorem2.3 (iii), (ii), and Theorem 2.4 (i’) that 1 = limn→∞

∑∞k=0ank =

∑∞k=0( lim

n→∞ank) = 0, a

contradiction. ¤

Example 2.7. Weak and strong convergence coincide in `1.

Proof. We assume that the sequence (x(n))∞n=0 is weakly convergent to x in `1, that isf(x(n)) − f(x) → 0 (n → ∞) for every f ∈ `∗1. Since `∗1 and `∞ are norm isomorphic,to every f ∈ `∗1 there corresponds a sequence a ∈ `∞ such that f(y) =

∑∞k=0akyk for

all y ∈ `1. We define the matrix B = (bnk)∞n,k=0 by bnk = x

(n)k − xk (n, k = 0, 1, . . . ).

Then we have f(x(n)) − f(x) =∑∞

k=0ak(x(n)k − xk) =

∑∞k=0bnkak → 0 (n → ∞) for all

a ∈ `∞, that is B ∈ (`∞, c0), and it follows from Theorem 2.4 (b) that ‖x(n) − x‖1 =∑∞k=0|x(n)

k − xk| =∑∞

k=0|bnk| → 0 (n →∞). ¤

Further results on matrix transformations and references can be found in [Boo, Coo,K–G, S–T, Z–B, Wil2, Mad, M–R, Mal, J–M], and in [Mad1] for infinite matrices ofoperators.


3. FK, BK, AK and AD Spaces

The theory of FK spaces is the most powerful tool in the theory of matrix transforma-tions ([Wil1, Wil2, K–G, Zel, M–R]). The fundamental result of this section is Theorem3.8 which states that matrix maps between FK spaces are continuous.

We start with a more general definition.

Definition 3.1. Let H be a linear space and a Hausdorff space. An FH space is aFrechet space, that is a locally convex linear metric space X, such that X is a subspaceof H and the topology of X is stronger than the restriction of the topology of H on X.If H = ω with its topology given by the metric d of (1.1), then an FH space is called anFK space.A BH space or a BK space is an FH or FK space which is a Banach space.

Remark 3.2. (a) If X is an FH space, then the inclusion map ι : X → H with ι(x) = xfor all x ∈ X is continuous. Therefore X is continuously embedded in H.(b) Since convergence in (ω, d) and coordinatewise convergence are equivalent ([Wil1,Theorem 4.1.1, p. 54]), convergence in an FK space implies coordinatewise convergence.(c) The letters F, H, K and B stand for Frechet, Hausdorff, Koordintate, the Germanword for coordinate, and Banach.

Example 3.3. Let H = F = {f : [0, 1] → IR}, and, for every t ∈ [0, 1], let t : F → IRbe the function with t(f) = t(f). We assume that F has the weak topology by Φ ={t : t ∈ [0, 1]}. Then C[0, 1] = {f ∈ F : f is continuous} is a BH space with ‖f‖ =supt∈[0,1] |f(t)|.Proof. Let (fk)

∞k=0 be a sequence in C[0, 1] with fk → 0 (k →∞), then t(fk) = fk(t) → 0

(k →∞) for all t ∈ Φ, that is fk → 0 (k →∞) in F . ¤

Example 3.4. Trivially ω is an FK space with the metric of (1.1). The spaces `∞, cand c0 and `p (1 ≤ p < ∞) are BK spaces with their natural norms, since |xk| ≤ ‖x‖ ineach case.

The following results are fundamental.

Theorem 3.5. ([Wil2, Theorem 4.2.2, p. 56]) Let X be a Frechet space, Y be an FHspace and f : X → Y be linear. Then f : X → H is continuous, if and only if f : X → Yis continuous.

Proof. Let TX , TY and TH be the topologies on X, Y and of H on Y , respectively.First, we assume that f : (X,Y ) is continuous. Since Y is an FH space, we haveTH ⊂ TY , and so f : X → H is continuous.Conversely, we assume that f : X → (Y, TH) is continuous, then it has closed graph bythe closed graph lemma (Theorem A.1). Since Y is an FH space, we again have TH ⊂ TY ,and so f : X → (Y, TY ) has closed graph. Hence f : X → (Y, TY ) is continuous by theclosed graph theorem (Theorem A.2). ¤

We obtain as an immediate consequence of Theorem 3.5.


Corollary 3.6. ([Wil2, Corollary 4.2.3, p. 56]) Let X be a Frechet space, Y be an FKspace, f : X → Y be linear, and Pn : X → |C (n = 0, 1, . . . ) be defined by Pn(x) = xn forall x ∈ X. If Pn ◦ f : X → |C is continuous for every n, then f : X → Y is continuous.

Proof. Since convergence and coordinatewise convergence are equivalent in ω by Remark3.2 (b), the continuity of Pn : X → |C for all n implies the continuity of f : X → ω,hence of f : X → Y by Theorem 3.5. ¤

By φ we denote the set of all finite sequences. Thus x = (xk)∞k=0 ∈ φ if and only if

there is an integer k such that xj = 0 for all j > k.

Theorem 3.7. ([M–R, Remark 1.16, p. 152]) Let X ⊃ φ be an FK space. If theseries

∑∞k=0akxk converge for all x ∈ X, then the linear functional fa defined by fa(x) =∑∞

k=0akxk for all x ∈ X is continuous.

Proof. We define the functionals fa (n ∈ IN0) by f[n]a (x) =

∑nk=0 akxk for all x ∈ X.

Since X is an FK space and f[n]a is a finite linear combination of coordinates, we have

f[n]a ∈ X ′ for all n. By hypothesis, the limits fa(x) = limn→∞ f

[n]a (x) exist for all x ∈ X,

hence fa ∈ X ′ by the Banach–Steinhaus theorem (Theorem A.3). ¤Theorem 3.8. ([Wil2, Theorem 4.2.8, p. 57]) Any matrix map between FK spaces iscontinuous.

Proof. Let X and Y be FK spaces, A ∈ (X, Y ) and fA : X → Y be defined by fA(x) =Ax for all x ∈ X. Since the maps Pn ◦ fA : X → |C are continuous for all n by Theorem3.7, fA : X → Y is continous by Corollary 3.6. ¤

It turns out that the FH topology of an FH space is unique.

Theorem 3.9. ([Wil2, Corollary 4.2.4, p. 56]) Let X and Y be FH spaces with X ⊂ Y .Then the topology TX is larger than the topology TY |X of Y on X.They are equal if and only if X is a closed subspace of Y .In particular, the topology of an FH space is unique.

Proof. Since X is an FH space, the inclusion map ι : X → H is continuous by Remark3.2 (a), hence ι : X → Y is continuous by Theorem 3.5. This implies TX ⊃ TY |X .Now let T and T ′ be FH topologies for an FK space. Then it follows by what we havejust shown that T ⊂ T ′ ⊂ T .If X is closed in Y , then X becomes an FH space with TY |X . It follows from theuniqueness that TX = TY |X .If TX = TY |X , then X is a complete, hence closed, subspace of Y . ¤

The class of FK spaces is fairly large.

Example 3.10. A Banach sequence space which is not a BK spaceWe consider the spaces (c0, ‖ · ‖∞) and (`2, ‖ · ‖2). Since they have the same algebraicdimension, there is an isomorphism f : c0 → `2. We define a second norm ‖ · ‖ on c0

by ‖x‖ = ‖f(x)‖2 for all x ∈ c0. Then (c0, ‖ · ‖) becomes a Banach space. But c0 and`2 are not linearly homeomorphic, since `2 is reflexive, and c0 is not. Therefore the two


norms on c0 are incomparable. By Example 3.4 and Theorem 3.9, (c0, ‖ · ‖) is a Banachsequence space which is not a BK space.

Theorem 3.11. ([Wil2, Theorem 4.2.5, p. 57]) Let X, Y and Z be FH spaces withX ⊂ Y ⊂ Z. If X is closed in Z, then X is closed in Y .

Proof. Since X is closed in (Y, TZ |Y ), it is closed in (Y, TY ) by Theorem 3.9. ¤Let Y be a topological space, and E ⊂ Y . Then we write clY (E) for the closure of E

in Y .

Theorem 3.12. ([Wil2, Theorem 4.2.7, p. 57]) Let X and Y be FH spaces with X ⊂ Y ,and E be a subset of X. Then we have

clY (E) = clY (clX(E)), in particular clX(E) ⊂ clY (E).

Proof. Since TY |X ⊂ TX by Theorem 3.9, it follows that clX(E) ⊂ clY (E). This implies

clY (clX(E)) ⊂ clY (clY (E)) = clY (E).

Conversely, E ⊂ clX(E) implies clY (E) ⊂ clY (clX(E)). ¤Example 3.13. (a) Since c0 and c are closed in `∞, their BK topologies are the same;since `1 is not closed in `∞, its BK topology is strictly stronger than that of `∞ on `1

(Theorem 3.9).(b) If c is not closed in an FK space X, then X must contain unbounded sequences(Theorem 3.11).

Definition 3.14. Let X ⊃ φ be an FK space. Then X is said to have(a) AD if clX(φ) = X;(b) AK if every sequence x = (xk)

∞k=0 ∈ X has a unique representation x =

∑∞k=0xke

(k).

Remark 3.15. The letters A, D and K stand for abschnittsdicht, the German word forsectionally dense, and Abschnittskonvergenz, the German word for sectional convergence.

Example 3.16. (a) Every FK space with AK obviously has AD.(b) An Example of an FK space with AD which does not have AK can be found in[Wil2, Example 5.2.14, p. 80].(c) The spaces ω, c0 and `p (1 ≤ p < ∞) have AK.(d) The space c does not have AK; every sequence x = (xk)

∞k=0 ∈ c has a unique repre-

sentation x = ξ e +∑∞

k=0(xk − ξ)e(k) where ξ = limk→∞ xk.(e) The space `∞ has no Schauder basis, since it is not separable.

Theorem 3.17. ([Wil2, 8.3.6, p. 123]) Let X be an FK space with AD, and Y and Y1

be FK spaces with Y1 a closed subspace of Y . Then A ∈ (X, Y1) if and only if A ∈ (X, Y )and Ae(k) ∈ Y1 for all k.

Proof. First, we assume A ∈ (X, Y1). Then Y1 ⊂ Y implies A ∈ (X,Y ), and e(k) ∈ X forall k implies Ae(k) ∈ Y1 for all k.Conversely, we assume A ∈ (X,Y ) and Ae(k) ∈ Y1 for all k. We define the map fA : X →Y by fA(x) = Ax for all x ∈ X. Then Ae(k) ∈ Y1 implies fA(φ) ⊂ Y1. By Theorem 3.8,


fA is continuous, hence fA(clX(φ)) = clY (fA(φ)). Since Y1 is closed in Y , and φ is densein the AD space X, we have fA(X) = fA(clX(φ)) = clY (fA(φ)) ⊂ clY (Y1) = clY1(Y1) = Y1

by Theorem 3.9. ¤

Theorem 3.18. ([Wil2, 8.3.7, p. 123]) Let X be an FK space, X1 = X ⊕ e = {x1 =x + λe : x ∈ X, λ ∈ |C}, and Y be a linear subspace of ω. Then A ∈ (X1, Y ) if and onlyif A ∈ (X,Y ) and Ae ∈ Y .

Proof. First, we assume A ∈ (X1, Y ). Then X ⊂ X1 implies A ∈ (X,Y ), and e ∈ X1

implies Ae ∈ Y .Conversely, we assume A ∈ (X,Y ) and Ae ∈ Y . Let x1 ∈ X1 be given. Then thereare x ∈ X and λ ∈ |C such that x1 = x + λe, and it follows that Ax1 = A(x + λe) =Ax + λAe ∈ Y . ¤

We close this section with two applications of our results.Let (X, d) be a metric space, δ > 0 and x0 ∈ X. Then we write Bδ[x0] = {x ∈ X :

d(x, x0) ≤ δ} for the closed ball of radius δ with its centre in x0. If X ⊂ ω is a linearmetric space and a ∈ ω, then we write

‖a‖∗δ = ‖a‖∗X,δ = supx∈Bδ[0]

∣∣∣∣∣∞∑

k=0

akxk

∣∣∣∣∣ ,

provided the expression on the right hand exists and is finite which is the case wheneverthe series

∑∞k=0akxk converge for all x ∈ X (Theorem 3.7); if X is a normed space then

we write

‖a‖∗ = ‖a‖X∗ = supx∈SX

∣∣∣∣∣∞∑

k=0

akxk

∣∣∣∣∣ .

The first result is the characterisation of the class (X, `∞) for arbitrary FK spaces X.

Theorem 3.19. ([M–R, Theorem 1.23 (b)]) Let X be an FK space. Then we haveA ∈ (X, `∞) if and only if

(3.1) ‖A‖∗δ = supn‖An‖∗δ < ∞ for some δ > 0,

where An = (ank)∞k=0 denotes the sequence in the n–th row of the matrix A.

Proof. First, we assume that (3.1) is satisfied. Then the series Anx converge for allx ∈ Bδ[0] and for all n, and Ax ∈ `∞ for all x ∈ Bδ[0]. Since the set Bδ[0] is absorbingby [Wil1, Fact (ix), p. 53], we conclude that the series Anx converge for all n and allx ∈ X, and Ax ∈ `∞ for all x.Conversely, we assume A ∈ (X, `∞). Then the map LA : X → `∞ defined by

(3.2) LA(x) = Ax for all x ∈ X

is continuous by Theorem 3.8. Hence there exist a neighbourhood N of 0 in X and areal δ > 0 such that Bδ[0] ⊂ N and ‖LA(x)‖∞ < 1 for all x ∈ X. This implies (3.1). ¤


Theorem 3.20. ([M–R, Theorem 1.23. p. 155]) Let X and Y be BK spaces.(a) Then (X, Y ) ⊂ B(X,Y ), that is every A ∈ (X,Y ) defines an operator LA ∈ B(X, Y )by (3.2).(b) If X has AK then B(X,Y ) ⊂ (X, Y ).(c) We have A ∈ (X, `∞) if and only if

(3.3) ‖A‖(X,`∞) = supn‖An‖∗X < ∞;

if A ∈ (X, `∞) then

(3.4) ‖LA‖ = ‖A‖(X,`∞).

Proof. (a) This is Theorem 3.8.(b) Let L ∈ B(X, Y ) be given. We write Ln = Pn ◦ L for all n, and put ank =

Ln(e(k)) for all n and k. Let x = (xk)∞k=0 ∈ X be given. Since X has AK, we have

x =∑∞

k=0xke(k), and since Y is a BK space, it follows that Ln ∈ X∗ for all n. Hence we

obtain Ln(x) =∑∞

k=0xkLn(e(k)) =∑∞

k=0ankxk = (Ax)n for all n, and so L(x) = Ax.(c) This follows immediately from Theorem 3.19 and the definition of ‖A‖(X,`∞). ¤

4. Multiplier and Dual Spaces

The so–called β–duals are of greater interest than the continuous duals in the the-ory of matrix transformations. They naturally arise in the characterisations of matrixtransformations in connection with the convergence of the series Anx.

The β–duals of sequence spaces are special cases of multiplier spaces.

Definition 4.1. Let X and Y be subsets of ω. The set M(X, Y ) = {a ∈ ω : ax =(akxk)

∞k=0 ∈ Y for all x ∈ X} is called the multiplier space of X in Y . Special cases are

Xα = M(X, `1), Xβ = M(X, cs) and Xγ = M(X, bs), the α–, β and γ– duals of X.

Proposition 4.2. ([M–R, Lemma 1.25, p. 156]) Let X, X1, Y , Y1 ⊂ ω and {Xδ} be acollection of subsets of ω. Then we have

(i) Y ⊂ Y1 implies M(X, Y ) ⊂ M(X, Y1)(ii) X ⊂ X1 implies M(X1, Y ) ⊂ M(X,Y )(iii) X ⊂ M(M(X, Y ), Y )(iv) M(X, Y ) = M (M(M(X, Y ), Y ), Y )

(v) M

(⋃δ

Xδ, Y

)=

⋂δ

M(Xδ, Y ).

Proof. (i), (ii) Parts (i) and (ii) are trivial.(iii) If x ∈ X, then ax ∈ Y for all a ∈ M(X, Y ), and consequently x ∈ M(M(X, Y ), Y ).(iv) We replace X by M(X, Y ) in (iii) to obtain M(X, Y ) ⊂ M(M(M(X,Y ), Y ), Y ).

Conversely we have X ⊂ M(M(X,Y )) by (iii), so M(M(M(X, Y ), Y ), Y ) ⊂ M(X, Y )by (ii).

(v) First Xδ ⊂⋃

δ Xδ for all δ implies M(⋃

δ Xδ, Y ) ⊂ ⋂δ M(Xδ, Y ) by (ii).

Conversely, if a ∈ ⋂δ M(Xδ, Y ), then a ∈ M(Xδ, Y ) for all δ, and so we have ax ∈ Y for

all x ∈ Xδ and all δ. This implies ax ∈ Y for all x ∈ ⋃δ Xδ, hence a ∈ M(

⋃δ Xδ, Y ). ¤


Example 4.3. We have (i) M(c0, c) = `∞; (ii) M(c, c) = c; (iii) M(`∞, c) = c0.

Proof. (i) If a ∈ `∞, then ax ∈ C for all x ∈ c0, and so `∞ ⊂ M(c0, c).Conversely, we assume a 6∈ `∞. Then there is a subsequence ak(j) of the sequence a suchthat |ak(j)| > j + 1 for all j = 0, 1, . . . . We define the sequence x by

(4.1) xk =

(−1)j

ak(j)

(k = k(j))

0 (k 6= k(j))

(j = 0, 1, . . . ).

Then we have x ∈ c0 and ak(j)xk(j) = (−1)j for all j = 0, 1, . . . , hence ax 6∈ c. This showsM(c0, c) ⊂ `∞.

(ii) If a ∈ c, then ax ∈ c for all x ∈ c, and so c ⊂ M(c, c).Conversely, we assume a 6∈ c. Since e ∈ c and ae = a 6∈ c, we have a 6∈ M(c, c). Thisshows M(c, c) ⊂ c.

(iii) If a ∈ c0 then ax ∈ c for all x ∈ `∞, and so c0 ⊂ M(`∞, c0).Conversely, we assume a 6∈ c0. Then there are a real b > 0 and a subsequence (ak(j))

∞j=0

of the sequence a such that |ak(j)| > b for all j = 0, 1, . . . . We define the sequence xas in (4.1). Then we have x ∈ `∞ and ak(j)xk(j) = (−1)j for all j = 0, 1, . . . , hencea 6∈ M(`∞, c). This shows M(`∞, c) = c0. ¤Example 4.4. Let † denote any of the symbols α, β or γ. Then we have ω† = φ, φ† = ω,c†0 = c† = `†∞ = `1, `†1 = `∞, and `†p = `q (1 < p < ∞; q = p/(p− 1)).

Another dual space frequently arises in the theory of sequence spaces.

Definition 4.5. Let X ⊃ φ be an FK space. Then Xf = {(f(e(n)))∞n=0 : f ∈ X ′} iscalled the functional dual of X.

Theorem 4.6. (a) We have Xα ⊂ Xβ ⊂ Xγ and X ⊂ X†† where † is any of the symbolsα, β and γ.(b) Let X ⊃ φ be an FK space. Then we have Xf = (clX(φ))f ([Wil2, Theorem 7.2.4,p. 106]).(c) Let X, Y ⊃ φ be FK spaces. If X ⊂ Y then Xf ⊃ Y f . If X is closed in Y thenXf = Y f ([Wil2, Theorem 7.2.4, p. 106]).

Proof. (a) Since `1 ⊂ cs ⊂ bs, it follows from Proposition 4.2 (i) that Xα ⊂ Xβ ⊂ Xγ,and Proposition 4.2 (iii) yields X ⊂ X††.

(b) We write Z = clX(φ).First, we assume that a ∈ Xf , that is an = f(e(n)) (n = 0, 1, . . . ) for some f ∈ X ′. Wewrite g = f |Z for the restriction of f to Z. Then an = g(e(n)) for all n = 0, 1, . . . , g ∈ Z ′

and so a ∈ Zf .Conversely, let a ∈ Z, then an = g(e(n)) (n = 0, 1, . . . ) for some g ∈ Z ′. By the Hahn–Banach–Theorem (Theorem A.4), g can be extended to f ∈ X ′, and we have an = f(e(n))for n = 0, 1, . . . , hence a ∈ Xf .

(c) We assume that a ∈ Y f . Then an = f(e(n)) (n = 0, 1, . . . ) for some f ∈ Y ′.Since X ⊂ Y , we have g = f |X ∈ X ′ by Theorem 3.9. If X is closed in Y , then the FK


topologies are the same by Theorem 3.9, and we obtain Xf = (clX(φ))f = (clY (φ))f = Y f

from Part (b). ¤It might be expected from X ⊂ X†† that X is contained in Xff ; but this is not the

case in general (Example 4.7). We will, however, see below that X ⊂ Xff for BK spaceswith AD (Theorem 4.16).

Example 4.7. Let X = c0 ⊕ z with z unbounded. Then X is a BK space, Xf = `1 andXff = `∞, so X 6⊂ Xff .

Theorem 4.8. ([Wil2, Theorem 7.2.7, p. 106]) Let X ⊃ φ be an FK space.(a) We have Xγ ⊂ Xf .(b) If X has AK, then Xβ = Xf .(c) If X has AD then Xβ = Xγ.

Proof. Let a ∈ Xβ. We define the linear functional f by f(x) =∑∞

k=0akxk for all x ∈ X.Then f ∈ X ′ by Theorem 3.7, and we have f(e(n)) = an for all n, hence a ∈ Xf . Thuswe have shown

(4.2) Xβ ⊂ Xf .

(b) Now we assume that X has AK, and a ∈ Xf . Let x ∈ X be given. Thenx =

∑∞k=0xke

(k), since X has AK, and since f ∈ X ′, we have f(x) = f(∑∞

k=0xke(k)) =∑∞

k=0xkf(e(k)) =∑∞

k=0xkak, hence a ∈ Xβ. Thus we have shown Xf ⊂ Xβ. Togetherwith Xβ ⊂ Xf , this yields Xβ = Xf .

(c) Now we assume that X has AD and a ∈ Xγ. We define the linear functionalsfn for n = 0, 1, . . . by fn(x) =

∑nk=0 akxk (x ∈ X). Since X is an FK space, we have

fn ∈ X ′ for all n. Furthermore, a ∈ Xγ implies that the sequence (fn)∞n=0 is pointwisebounded, hence equicontinuous by the uniform boundedness principle (Theorem A.5).Since limn→∞ fn(x) exists for all x ∈ X and X has AD, it must exists for all x ∈ X bythe convergence lemma (Theorem A.6), hence a ∈ Xβ. Thus we have shown Xγ ⊂ Xβ.We also have Xβ ⊂ Xγ by Theorem 4.6 (a), hence Xβ = Xγ.

(a) First we observe that clX(φ) ⊂ X implies Xγ ⊂ (clX(φ))γ by Proposition 4.2 (ii).Furthermore, we have (clX(φ))γ = (clX(φ))β ⊂ (clX(φ))f = Xf by Part (c), (4.2) andTheorem 4.6 (b). Thus we have shown Xγ ⊂ Xf . ¤

Now we establish a relationship between the β– and continuous duals of an FK space.

Theorem 4.9. ([Wil2, Theorem 7.2.9, p. 107]) Let X ⊃ φ be an FK space. ThenXβ ⊂ X ′; this means, that there is a linear one–to–one map T : Xβ → X ′. If X hasAK then T is onto.

Proof. We define the map T by Ta = fa (a ∈ Xβ) where fa is the functional withfa =

∑∞k=0akxk for all x ∈ X, and observe that Ta = fa ∈ X ′ for all a ∈ Xβ by Theorem

3.7. Obviously T is linear. Furthermore, if Ta = 0 then fa(x) =∑∞

k=0akxk = 0 for allx ∈ X, in particular fa(e

(k)) = ak = 0 for all k, that is a = 0. Thus Ta = 0 impliesa = 0, and consequently T is one–to–one.Now we assume that X has AK. Let f ∈ X ′ be given. We define the sequence a by


ak = f(e(k)) for k = 0, 1, . . . . Let x ∈ X be given. Then x =∑∞

k=0xke(k), since X has

AK, and f ∈ X ′ implies f(x) = f(∑∞

k=0xke(k)) =

∑∞k=0xkf(e(k)) =

∑∞k=0xkak, hence

a ∈ Xβ and Ta = f . This shows that the map T is onto. ¤

A relation between the functional and continuous duals of an FK space is given by

Theorem 4.10. Let X ⊃ φ be an FK space.(a) Then the map q : X ′ → Xf given by q(f) = (f(e(k))∞k=0 is onto. Moreover, ifT : Xβ → X ′ denotes the map of Theorem 4.9, then q(Ta) = a for all a ∈ Xβ ([Wil2,Theorem 7.2.10, p. 107]).(b) Then Xf = X ′, that is the map q of Part (a) is one–to–one, if and only if X hasAD ([Wil2, Theorem 1.11.12, p. 108]).

Proof. (a) Let a ∈ Xf be given. Then there is f ∈ X ′ such that ak = f(e(k)) for all k,and so q(f) = (f(e(k))∞k=0 = a. This shows that q is onto.Now let a ∈ Xβ be given. We put f = Ta ∈ X ′ and obtain q(Ta) = q(f) = (f(e(k)))∞k=0 =((Ta)(e(k)))∞k=0 = (ak)

∞k=0 = a.

(b) First we assume that X has AD. Then q(f) = 0 implies f = 0 on φ, hence f = 0,since X has AD. This shows that q is one–to–one.Conversely we assume that X does not have AD. By the Hahn–Banach theorem, (The-orem A.4) there exists an f ∈ X ′ with f 6= 0 and f = 0 on φ. Then we have q(f) = 0,and q is not one–to–one. ¤

Example 4.11. We have cβ = cf = `1. The map T of Theorem 4.9 is not onto. Weconsider lim ∈ X ′. If there were a ∈ Xf with lim a =

∑∞k=0akxk then it would follow

that ak = lim e(k) = 0, hence lim x = 0 for all x ∈ c, contradicting lim e = 1. Also thenmap q of Theorem 4.10 is not onto, since q(lim) = 0.

It turns out that the multiplier spaces and the functional duals of BK spaces are againBK spaces. These results do not extend to FK spaces, in general.

Theorem 4.12. ([M–R, Theorem 1.30, p. 158]) Let X ⊃ φ and Y be BK spaces. ThenZ = M(X, Y ) is a BK space with ‖z‖ = supx∈SX

‖xz‖ for z ∈ Z.

Proof. Let ‖ · ‖X and ‖ · ‖Y denote the BK norms of X and Y .Every z ∈ Z defines a diagonal matrix map z : X → Y where z(x) = xz = (xkzk)

∞k=0

for all x ∈ X, and z ∈ B(X,Y ) by Theorem 3.20 (a). This embeds Z in B(X, Y ), for ifz = 0 then (z(en))n = zn = 0 for all n, hence z = 0.To see that the coordinates are continuous, we fix n and put u = 1/‖e(n)‖X andv = ‖e(n)‖Y . Then we have ‖ue(n)‖X = 1 and uv|zn| = u‖zne

(n)‖Y = u‖e(n)z‖Y =‖(ue(n))z‖Y ≤ ‖z‖.It remains to show that Z is a closed subspace of the Banach space B(X,Y ). Let(z(m))∞m=0 be a sequence in B(X, Y ) with z(m) → T ∈ B(X,Y ) (m → ∞). For everyfixed x ∈ X, we obtain z(m)(x) → T (x) ∈ Y (m → ∞), and since Y is a BK space,

this implies xkz(m)k = (z(m)(x))k → (T (x))k (m → ∞) for every fixed k. If we choose

x = e(k) then we obtain z(m)k → tk = (T (e(k)))k. Thus we have xkz

(m)k → xktk and


xkz(m)k → (T (x))k (m → ∞), hence T (x) = xt, and so T = t. This shows that Z is

closed. ¤

We obtain as an immediate consequence of Theorem 4.12.

Corollary 4.13. ([M–R, Corollary 1.31, p. 158]) The α–, β– and γ–duals of a BKspace X are BK spaces with ‖a‖α = supx∈SX

‖ax‖1 = supx∈SX(∑∞

k=0|akxk|) for all

a ∈ Xα, and ‖a‖β = supx∈SX‖a‖bs = supx∈SX

(supn |∑n

k=0 akxk|) for all a ∈ Xβ, Xγ.

Furthermore, Xβ is a closed subspace of Xγ.

Proof. The first part is an immediate consequence of Theorem 4.12.Since the BK norms on Xβ and Xγ are the same and Xβ ⊂ Xγ by Theorem 4.6, thesecond part follows from Theorem 3.9. ¤

Theorem 4.12 fails to hold for FK spaces, in general.

Example 4.14. The space ω is an FK space, and ωα = ωβ = ωγ = φ, but φ has noFrechet metric.

We give the following result without proof.

Theorem 4.15. ([Wil2, Theorem 7.2.14, p. 108]) Let X ⊃ φ be a BK space. Then Xf

is a BK space.

Theorem 4.16. ([Wil2, Theorem 7.2.15, p. 108]) Let X ⊃ φ be a BK space. ThenXff ⊃ clX(φ). Hence, if X has AD, then X ⊂ Xff .

Proof. First we have to show φ ⊂ Xf in order for Xff to be meaningful.This is true because Pk ∈ X ′ for all k where Pk(x) = xk (x ∈ X) since X is a BK space,and q(Pk) = e(k) (Theorem 4.10 (a)).Since the second part is equivalent to the first part by Theorem 4.6 (b), we assume thatX has AD, and have to show X ⊂ Xff .Let x ∈ X be given. We define the functional f : X ′ → |C by f(ψ) = ψ(x) for all ψ ∈ X ′.Then we have |f(ψ)| = |ψ(x)| ≤ ‖ψ‖‖x‖, and consequently f ∈ X ′′. Let q : X ′ → Xf

be the map of Theorem 4.10 (a) which is an isomorphism by Theorem 4.10 (b), since Xhas AK. Thus the inverse map q−1 : Xf → X ′ exists. We define the map g : Xf → |Cby g(b) = ψ(x) (b ∈ Xf ) where x = q−1(b). It follows that

|g(b)| = |ψ(x)| = |f(ψ)| ≤ ‖f‖ ‖ψ‖ = ‖f‖ ‖q−1(b)‖ ≤ ‖f‖ ‖q−1‖ ‖b‖,and the open mapping theorem (Theorem A.7) yields ‖q−1‖ < ∞. Thus we have g ∈(Xf )′. Finally it follows that xk = Pk(x) = g(q(Pk)) = g(e(k)) for all k, hence x ∈ Xff .Thus we have shown X ⊂ Xff . ¤

The condition that X has AD is not necessary for X ⊂ Xff , in general.

Example 4.17. Let X = c0 ⊕ z with z ∈ `∞. Then we have Xff = `f1 = `∞ ⊃ X, but X

does not have AD.


5. Matrix Transformations

We apply the results of the previous sections to give necessary and sufficient conditionson the entries of a matrix A to be in a class (X, Y ).

The first two results concern the transpose AT of a matrix A.

Theorem 5.1. ([Wil2, Theorem 8.3.8, p. 124]) Let X be an FK space and Y be anyset of sequences. If A ∈ (X,Y ) then AT ∈ (Y β, Xf ). If X and Y are BK spaces andY β has AD then we have AT ∈ (Y β, clXf (Xβ)).

Proof. Let A ∈ (X,Y ) and z ∈ Y β be given. We define the functional f : X → |Cby f(x) =

∑∞n=0 znAnx (x ∈ X). Since X is an FK space, Ax ∈ Y by assumption

and z ∈ Y β, we have f ∈ X ′ by Theorem 3.7. Furthermore it follows that f(e(k)) =∑∞n=0 znank = AT

k z for all k, hence AT z ∈ Xf . This shows that AT ∈ (Y β, Xf ).Now we assume that X and Y are BK spaces and Y has AD. Then Xβ ⊂ Xf byTheorems 4.6 (a) and 4.8 (a), and Xf is a BK space by Theorem 4.15. Also clXf (Xβ)is a closed subspace of Xf . Since A ∈ (X, Y ), we have An = (ank)

∞k=0 ∈ Xβ for all n,

but AT e(n) = (∑∞

j=0 ajke(n)j )∞k=0 = (ank)

∞k=0 = An for all n. So we have AT e(n) ∈ Xβ for

all n, and this and A ∈ (Y β, Xf ) imply AT ∈ (Y β, clXf (Xβ)) by Theorem 3.17. ¤

Theorem 5.2. ([Wil2, Theoren 8.3.9, p. 124]) Let X and Z be BK spaces with AKand Y = Zβ. Then we have (X, Y ) = (Xββ, Y ); furthermore A ∈ (X, Y ) if and only ifAT ∈ (Z,Xβ).

Proof. Since X is a BK space with AK, Xβ is a BK space by Corollary 4.13, andXβ = Xf by Theorem 4.8 (b).First we assume A ∈ (X,Y ). Then it follows by Theorem 5.1 and since Zββ ⊃ Z byTheorem 4.6 (a) that AT ∈ (Y β, Xβ) = (Zββ, Xβ) ⊂ (Z,Xβ).Conversely, if AT ∈ (Z, Xβ) then it follows by Theorem 5.1 and since Xββ ⊃ X byTheorem 4.6 (a) that A ∈ (Xββ, Zβ) ⊂ (X,Zβ) = (X, Y ). This proves the second part.To prove the first part, we first observe that X ⊂ Xββ implies (Xββ, Y ) ⊂ (X, Y ).Conversely we assume A ∈ (X, Y ). Then we have AT ∈ (Z,Xβ) as proved above, andTheorem 5.1 implies A = ATT ∈ (Xββ, Zβ) = (Xββ, Y ). ¤

Remark 5.3. The results of the previous sections yield the characterisations of the classes(X, Y ) where X and Y are any of the spaces `p (1 ≤ p ≤ ∞), c0, c with the exceptionsof (`p, `r) where both p, r 6= 1,∞ (the characterisations are unknown), and of (`∞, c)(Schur’s theorem 2.4) and (`∞, c0) ([S–T, 21 (21.1)]) for which no functional analyticproofs seem to be known.

The class (`2, `2) was characterised by Crone ([Cro] or [Ruc, pp. 111–115]).

Example 5.4. (a) We have (c0, `∞) = (c, `∞) = (`∞, `∞); furthermore A ∈ (`∞, `∞) ifand only if

(5.1) ‖A‖(∞,∞) = supn

∞∑

k=0

|ank| < ∞.


If A is in any of the classes above then ‖LA‖ = ‖A‖(∞,∞).(b) We have A ∈ (c0, c) if and only if (5.1) holds and

(5.2) limn→∞

ank = αk exists for every k.

If A ∈ (c0, c) then

(5.3) limn→∞

An(x) =∞∑

k=0

αkxk.

(c) (Toeplitz’s theorem 2.3) We have A ∈ (c, c) if and only if (5.1) and (5.2) hold, and

(5.4) limn→∞

∞∑

k=0

ank = α exists.

If A ∈ (c, c) and x ∈ c then

(5.5) limn→∞

An(x) =

(α−

∞∑

k=0

αk

)limk→∞

xk +∞∑

k=0

αkxk.

Furthermore have A ∈ (c, c; P ) if and only if (5.1), (5.2) and (5.4) hold with αk = 0(k = 0, 1, . . . ) and α = 1.

Proof. (a) We have A ∈ (c0, `∞) if and only if (5.1) by (3.3) in Theorem 3.20, and since

cβ0 = `1 and c∗0 and `1 are norm isomorphic.

Furthermore c0 ⊂ c ⊂ `∞ implies (`∞, `∞) ⊂ (c, `∞) ⊂ (c0, `∞).

Also (c0, `∞) = (cββ0 , `∞) = (`∞, `∞) by the first part of Theorem 5.2.

The last part is obvious from Theorem 3.20.(b) Since c is a closed subspace of `∞, the characterisation of the class (c0, c) is an

immediate consequence of Theorem 3.17 and Part (a).Now we assume A ∈ (c0, c), and write ‖A‖ = ‖A‖(`∞,`∞), for short. Let m be a given non–negative integer Then it follows from (5.2) and (5.1) that

∑mk=0 |αk| = limn→∞

∑mk=0 |ank|

≤ ‖A‖. Since m was arbitrary, we have (αk)∞k=0 ∈ `1,

(5.6)∞∑

k=0

|αk| ≤ ‖A‖ and∞∑

k=0

|αkxk| < ‖A‖ ‖x‖∞ for all x ∈ c.

Now let x ∈ c0 and ε > 0 be given. Then we can choose an integer k(ε) such that|xk| ≤ ε/((4‖A‖+ 1)) for all k > k(ε), and by (5.2) we can choose and integer n(ε) such

that∑k(ε)

k=0 |ank − αk| |xk| < ε/2 for all n > n(ε). Let n > n(ε). Then (5.1) and (5.6)imply

∣∣∣∣∣Anx−∞∑

k=0

αkxk

∣∣∣∣∣ ≤k(ε)∑

k=0

|ank − αk|+∞∑

k=k(ε)

(|ank + αk|)|xk|

<ε

2+

ε

4‖A‖+ 1

( ∞∑

k=0

|ank|+∞∑

k=0

|αk|)≤ ε

2+

ε

2= ε.


Thus we have proved (5.3).(c) The characterisation of the class (c, c) is an immediate consequence of Part (a),

and Theorems 3.17 and 3.18.Now we assume A ∈ (c, c). Let x ∈ c be given and ξ = limk→∞ xk. Then x− ξe ∈ c0 andit follows from (5.3) and (5.5) that

Anx = An(x− ξe) + ξAne →∞∑

k=0

αk(xk − ξ) + ξα = ξ(α−∞∑

k=0

ak) +∞∑

k=0

αkxk,

which is (5.5).Finally, the characterisation of the class (c, c; P ) is an immediate consequence of thecharacterisation of (c, c) and (5.5). ¤Example 5.5. We have (`1, `1) = B(`1, `1) and A ∈ (`1, `1) if and only if

(5.7) ‖A‖(1,1) = supk

∞∑n=0

|ank| < ∞.

If A ∈ (`1, `1) then

(5.8) ‖LA‖ = ‖A‖(`1,`1).

Proof. Since `1 has AK, Theorem 3.20 (b) yields the first part.We apply the second part of Theorem 5.2 with X = `1, Z = c0, BK spaces with AK,and Y = Zβ = `1 to obtain A ∈ (`1, `1) if and only if AT ∈ (`∞, `∞); by Example 5.4(a), this is the case if and only if (5.7) is satisfied.Furthermore, if A ∈ (`1, `1) then

‖LA(x)‖1 =∞∑

n=0

∣∣∣∣∣∞∑

k=0

ankxk

∣∣∣∣∣ ≤∞∑

k=0

∞∑n=0

|ankxk| ≤ ‖A‖(1,1)‖x‖1

implies ‖LA‖ ≤ ‖A‖(1,1). Also LA ∈ B(`1, `1) implies ‖LA(x)‖1 = ‖Ax‖1 ≤ ‖LA‖ ‖x‖1,

and it follows from ‖e(k)‖1 = 1 for all k that ‖A‖(1,1) = supk

∑∞n=0 |ank| = supk ‖LA(e(k))‖1 ≤

‖LA‖. ¤

6. Measures of Noncompactness

Now we find necessary and sufficient conditions for a matrix A ∈ (X,Y ) to definea compact operator LA. This can be achieved by applying the Hausdorff measure ofnoncompactness.

The first measure of noncompactness was defined and studied by Kuratowski ([Kur]),and later used by Darbo ([Dar]). The Hausdorff measure of noncompactness was intro-duced and studied by Goldenstein, Gohberg and Markus ([GGM]). Istratesku introducedand studied the Istratesku measure of noncompactness ([Ist]). The interested reader isreferred for measures on noncompactness to [AKP, B–G, Ist1, TBA, M–R].

We only consider the Hausdorff measure of noncompactness; it is the most suitableone for our purposes.

We recall of few standard notations and definitions.


The convex hull of a set S in a linear space X is the set

conv(S) =

{x =

n∑

k=0

λksk : λk > 0, sk ∈ S (k = 0, 1, . . . , n) andn∑

k=0

λk = 1

}

of (finite) convex linear combinations of S.Let (X, d) be a metric space, x0 ∈ X and r > 0. By B(x0, r) = {x ∈ X : d(x, x0) < r}

we denote the open ball of radius r, centred at x0. If M is a subset of X, then M denotesthe closure of M . A set in a metric space is said to be totally bounded, if for every ε > 0it can be covered by a finite number of open balls of radius ε. It is well–known that asubset M of a metric space is compact if and only if every sequence (xn) in M has aconvergent subsequence, and in this case the limit of the subsequence is in M . The setM is said to be relatively compact if the closure M of M is a compact set. If the set Mis relatively compact, then M is totally bounded. If the metric space is complete, thenthe set M is relatively compact if and only if it is totally bounded. It is easy to provethat a subset M of a metric space X is relatively compact if and only if every sequence(xn) in M has a convergent subsequence; in this case, the limit of the subsequence neednot be in M .

Let X and Y be infinite–dimensional complex Banach spaces. A linear operator Lfrom X to Y is called compact if the domain of L is all of X, and, for every boundedsequence (xn) in X, the sequence (L(xn)) has a convergent subsequence in Y .

Now we give the definition of the Hausdorff measure of compactness of bounded setsin a metric space.

Definition 6.1. Let (X, d) be a metric space and M denote the collection of boundedsubsets of X. The function χ : M→ [0,∞) with

χ(Q) = inf

{ε > 0 : Q ⊂

n⋃

k=0

B(xk, rk); xk ∈ X, rk < ε (n = 0, 1, 2 . . . )

}

is called Hausdorff measure of noncompactness; χ(Q) is called the Hausdorff measure ofnoncompactness of Q.

The Hausdorff measure of noncompactness has the following basic properties.

Proposition 6.2. ([M–R, Lemma 2.11, p. 168]) Let X be a metric space and Q,Q1,Q2

∈M. Then we have

χ(Q) = 0 if and only if Q is totally bounded,(i)

χ(Q) = χ(Q),(ii)

Q1 ⊂ Q2 implies χ(Q1) ≤ χ(Q2),(iii)

χ(Q1 ∪Q2) = max{χ(Q1), χ(Q2)},(iv)

χ(Q1 ∩Q2) ≤ min{χ(Q1), χ(Q2)}.(v)

Proof. (i), (iii) The statements in (i) and (iii) follow directly from Definition 6.1.


(ii) We have χ(Q) ≤ χ(Q) by (iii).Let ρ = χ(Q). Then, given ε > 0, there are n = n(ε) ∈ IN and xk ∈ X such thatQ ⊂ ⋃n

k=0 B(xk, ρ + ε/2), and it follows that

Q ⊂n⋃

k=0

B(xk, ρ + ε/2) =n⋃

k=0

B(xk, ρ + ε/2) ⊂n⋃

k=0

B(x0, ρ + ε).

Since ε > 0 was arbitrary, this implies χ(Q) ≤ ρ.(iv) It follows from (iii), that χ(Qj) ≤ χ(Q1 ∪Q2) for j = 1, 2, hence

(6.1) max{χ(Q1), χ(Q2)} ≤ χ(Q1 ∪Q2).

Now let ρ = max{χ(Q1), χ(Q2)} and ε > 0 be given. Then, by Definition 6.1, Q1 andQ2 can be covered by finite unions of open balls of radius ρ + ε. Obviously the union ofthese covers is a finite cover of Q1∪Q2. This implies χ(Q1∪Q2) ≤ ρ+ε, and since ε > 0was arbitrary, it follows that χ(Q1 ∪ χ(Q2) ≤ ρ. Now this and (6.1) together imply theequality in (iv).

(v) It follows from (iii) that χ(Q1 ∩ Q2) ≤ χ(Qj) for j = 1, 2, hence χ(Q1 ∩ Q2) ≤min{χ(Q1), χ(Q2)}. ¤

Proposition 6.3. ([M–R, Theorem 2.12, p. 169]) Let X be a normed space and Q,Q1,Q2

∈M. Then we have

χ(Q1 + Q2) ≤ χ(Q1) + χ(Q2),(i)

χ(Q + x) = χ(Q) for all x ∈ X,(ii)

χ(λQ) = |λ|χ(Q) for all scalars,(iii)

χ(Q) = χ(conv(Q)).(iv)

Proof. We denote the norm of X by ‖ · ‖.(i) Let ρj = χ(Qj) for j = 1, 2, ρ = ρ1 + ρ2, and ε > 0 be given. Then there are

nj = nj(ε) ∈ IN0 and x(j)k ∈ X (0 ≤ k ≤ nj) for j = 1, 2 such that

(6.2) Qj ⊂nj⋃

k=0

B(x(j)k , ρj + ε/2) for j = 1, 2.

Let x ∈ Q1 + Q2. Then there are xj ∈ Qj (j = 1, 2) such that x = x1 + x2, and it

follows from (6.2) that there are kj ∈ {0, 1, . . . , nj} such that xj ∈ B(x(j)kj

, ρj + ε/2) for

j = 1, 2. This implies ‖x − (x(1)k1

+ x(2)k2

)‖ ≤ ‖x1 − x(1)kj‖ + ‖x2 − x

(2)k2‖ < ρ + ε, and

so Q1 + Q2 ⊂⋃n1

k=0

⋃n2

j=0 B(x(1)k + x

(2)j , ρ + ε). Since ε > 0 was arbitrary, we conclude

χ(Q1 + Q2) ≤ ρ.(ii) Let x ∈ X. Since obviously χ({x}) = χ({−x}) = 0, it follows from (i) that

χ(Q) = χ ((Q + x)− x) ≤ χ(Q + x) + χ({−x})= χ(Q + x) ≤ χ(Q) + χ({x}) = χ(Q).


(iii) Since the equality in (iii) is trivial for λ = 0, we assume λ 6= 0. Let ρ = χ(Q) andε > 0 be given. Then we have

(6.3) Q ⊂n⋃

k=0

B(xk, ρ + ε)

Let y ∈ λQ be given. Then there are x ∈ Q such that y = λx, and k0 ∈ {0, 1, . . . , n}such that x ∈ B(xk, ρ + ε). We put yk = λxk for k = 0, 1, . . . and obtain ‖y − yk0‖ =|λ| ‖x − xk‖ < |λ|(ρ + ε). This implies λQ ⊂ ⋃∞

k=0 B(yk, |λ|(ρ + ε)). Since ε > 0 wasarbitrary, we conclude χ(λQ) ≤ |λ|ρ = |λ|χ(Q). Furthermore, it follows by what wehave just shown that χ(Q) = χ(λ−1(λQ)) ≤ |λ−1|χ(λQ), hence |λ|χ(Q) ≤ χ(λQ).

(iv) Since obviously Q ⊂ conv(Q), we obtain χ(Q) ≤ χ(conv(Q)).We have to show

(6.4) χ(conv(Q)) ≤ χ(Q).

Let ρ = χ(Q) and ε > 0 be given. Then we have (6.3), and every ball Bk = B(xk, ρ + ε)is a convex set. To see this, let x, y ∈ Bk and 0 ≤ λ ≤ 1. Then we have

‖λx + (1− λ)y − xk‖ ≤ ‖λ(x− xk)‖+ ‖(1− λ)(y − xk)‖ < (λ + (1− λ))(ρ + ε) < ρ + ε.

We define σ = {λ = (λ0, . . . , λn) ∈ IRn :∑n

k=0 λk = 1 and λk ≥ 0 for k = 0, . . . , n} andA(λ) =

∑nk=0 λkBk for every λ ∈ σ.

It follows from (i) and (iii) that

(6.5) χ(A(λ)) ≤n∑

k=0

λkχ(Bk) ≤ ρ + ε.

Now we show that the set A =⋃

λ∈σ A(λ) is convex.Let x, y ∈ A. Then there are λ, µ ∈ σ such that x ∈ A(λ) and y ∈ A(µ), hence x =∑n

k=0 λkxk and y =∑n

k=0 µkyk with λ = (λ0, . . . , λn), µ = (µ0, . . . , µn) and xk, yk ∈ Bk

(k = 0, 1, . . . ). We put z = tx+(1− t)y where 0 ≤ t ≤ 1 and η = tλ+(1− t)µ and haveto show z ∈ A(η) for some η ∈ σ. Putting ηk = tλk + (1− t)µk, ξk = tλk/ηk for ηk > 0and ξk = 0 for ηk = 0, and zk = ξkxk + (1− ξ)yk (k = 0, 1, . . . , n), we obtain

n∑

k=0

ηkzk =n∑

k=0

(ξkxk + (1− ξ)yk) =n∑

k=0

(tλkxk + (1− t)µkyk) = z.

Since each Bk is a convex set, we have zk ∈ Bk for k = 0, 1, . . . . Furthermore, weobviously have ηk ≥ 0 and

∑nk=0 ηk = t

∑nk=0 λk + (1− t)

∑nk=0 µk = 1, hence η ∈ σ and

so z ∈ A(η). Thus we have shown that A is convex.Now we can prove the result.Since Q ⊂ ⋃n

k=0 Bk ⊂ A and the set A is convex, it follows that conv(B) ⊂ A. Sincethe set σ is compact, given ε > 0, we can find finitely many λ(0), . . . , λ(m) ∈ σ suchthat for all λ ∈ σ we have mink=0,...,m{‖λ− λ(k)‖1} < ε/M where M = supk=0,...,n{‖x‖ :x ∈ Bk} < ∞. So if x ∈ A, x =

∑nk=0 λkxk, λk ≥ 0,

∑nk=0 λk = 1, then there exists

j ∈ {0, 1, . . . ,m} such that∑n

k=0 |λk−λ(j)k | < ε/M . We put x =

∑nk=0 λ

(j)k xk and obtain


‖x − x‖ ≤ ∑nk=0 |λk − λ

(j)k | ‖xk‖ < ε, and therefore conv(B) ⊂ ⋃m

j=0 A(λ(j)) + εB(0, 1).

Thus we have by Proposition 6.2 (iv) and (6.5)

χ(conv(B)) ≤ maxj=0,...,m

{χ(A(λ(j)) + χ(ε(B(0, 1))} ≤ ρ + ε + 2ε.

Since ε > 0 was arbitrary, it follows that (6.4) holds. ¤

Theorem 6.4. ([M–R, Theorem 2.13, p. 169]) Let X be an infinite–dimensional normedspace. Then χ(B(0, 1)) = 1.

Proof. We write B = B(0, 1).Obviously we have χ(B) ≤ 1. If χ(B) = ρ < 1 then we choose ε > 0 such thatρ + ε < 1. Then we have B ⊂ ⋃n

k=0 B(xk, ρ + ε) ⊂ ⋃nk=0(xk + (ρ + ε)B), and it follows

from Proposition 6.2 (iv) and Proposition 6.3 (ii) and (iii) that

ρ = χ(B) ≤ max0≤k≤n

χ (xk + (q + ε)B) = (ρ + ε)q.

Since q + ε < 1, this implies q = 0, and so B is a totally bounded set by Proposition6.2 (i). But this is impossible, since X is an infinite–dimensional space. Thus we haveχ(B) = 1. ¤

Theorem 6.5 (Goldenstein, Gohberg, Markus). ([GGM]; [M–R, Theorem 2.23, p. 173])Let X be a Banach space with a Schauder basis (bk)

∞k=0, Q ∈M and Pn : X → X be the

projector onto the linear span of {b0, b1, . . . , bn}. Then we have

(6.6)1

alim sup

n→∞

(supx∈Q

‖(I − Pn)(x)‖)≤ χ(Q) ≤ lim sup

n→∞

(supx∈Q

‖(I −Pn)(x)‖)

where a = lim supn→∞ ‖I − Pn‖.Proof. Obviously we have for every non–negative integer n

(6.7) Q ⊂ Pn(Q) + (I − Pn)(Q).

It follows from (6.7) and Propositions 6.2 and 6.3 that

(6.8) χ(Q) ≤ χ(Pn(Q)) + χ((I − Pn)(Q)) = χ((I − Pn)(Q)) ≤ supx∈Q

‖(I −Pn)(x)‖ ,

and we obtain

(6.9) χ(Q) ≤ infn

(supx∈Q

‖(I − Pn)(x)‖)≤ lim sup

n→∞

(supx∈Q

‖(I − Pn)(x)‖)

.

This proves the second inequality in (6.6).Now we show the first inequality in (6.6).Let ρ = χ(Q) and ε > 0 be given. Then we have

Q ⊂n⋃

k=0

Bk(xk, ρ + ε) ⊂ {x0, x1, . . . , xn}+ (ρ + ε)B(0, 1).


This implies that for every x ∈ Q there exist y ∈ {x1, x2, . . . , xn} and z ∈ B(0, 1) suchthat x = y + (ρ + ε)z, and so

supx∈Q

‖(I − Pn)(x)‖ ≤ sup0≤k≤n

‖(I − Pn)(xk)‖+ (ρ + ε)‖I − Pn‖.

This yields

lim supn→∞

(supx∈Q

‖(I − Pn)(x)‖)≤ (ρ + ε) lim sup

n→∞‖I − Pn‖.

Since ε > 0 was arbitrary, the first inequality in (6.6) follows. ¤

So far we considered the measure of noncompactness of bounded subsets of a metricspace. Now we define the measure of noncompactness of an operator.

Definition 6.6. Let κ1 and κ2 be measures of noncompactness on the Banach spacesX and Y , and MX and MY denote the collections of bounded sets in X and Y . Anoperator L : X → Y is said to be (κ1, κ2)–bounded if

L(Q) ∈MY for all Q ∈MX

and there exists a non–negative real c such that

κ2(L(Q)) ≤ c κ1(Q) for all Q ∈MX .

If an operator L is (κ1, κ2)–bounded, then the number

(6.10) ‖L‖(κ1,κ2) = inf{c ≥ 0 : κ2(L(Q)) ≤ c κ1(Q) for all Q ∈MX}is called the (κ1, κ2)–measure of noncompactness of L.If κ = κ1 = κ2, then we write ‖L‖κ = ‖L‖(κ,κ).

The following result is useful to compute the Hausdorff measure of noncompactnessof a bounded linear operator between Banach spaces.

Theorem 6.7. ([M–R, Theorem 2.25, p. 175]) Let X and Y be Banach spaces, L ∈B(X,Y ), SX and BX be the unit sphere and the closed unit ball in X, and χ be theHausdorff measure of noncompactness. Then we have

(6.11) ‖L‖χ = χ(L(SX)) = χ(L(BX)).

Proof. Since conv(S) = BX and L(conv(SX)) = conv(L(SX)), it follows from Proposition6.3 (iv) that

(6.12) χ(L(BX)) = χ(L(conv(SX))) = χ(conv(L(SX))) = χ(L(SX)),

and we have χ(L(BX)) ≤ ‖L‖χ χ(B) = ‖L‖χ by (6.10) and Theorem 6.4.To prove the converse inequality, let Q ∈ MX be given, ρ = χ(Q) and ε > 0 be given.Then we have Q ⊂ ⋃n

k=0 B(xk, ρ + ε) and obviously L(Q) ⊂ ⋃nk=0 L(B(xk, ρ + ε). It


follows from this, Proposition 6.2 (ii), Proposition 6.2 (ii),and (iii) that

χ(L(Q)) ≤ χ

(n⋃

k=0

L(B(xk, ρ + ε))

)≤ χ (L({x0, . . . , xn}+ B(0, ρ + ε))

= χ(L(B(0, ρ + ε))) ≤ χ(L((ρ + ε)B(0, 1))) = χ((ρ + ε)L(B(0, 1)))

≤ (ρ + ε)χ(L(B)).

Since ε > 0 was arbitrary, we have χ(L(Q)) ≤ ρχ(L(BX)) = χ(Q)χ(L(BX)) for allQ ∈MX , hence ‖L‖χ ≤ χ(L(BX)). ¤

Theorem 6.8. ([M–R, Corollary 2.26, p. 175]) Let X, Y and Z be Banach spaces,L ∈ B(X,Y ), L ∈ B(Y, Z), and C(X,Y ) denote the set of compact operators in B(X,Y ).Then ‖ · ‖χ is a seminorm on B(X,Y ) and

‖L‖χ = 0 if and only if L ∈ C(X,Y ),(6.13)

‖L‖χ ≤ ‖L‖,‖L ◦ L‖χ ≤ ‖L‖χ ‖L‖χ.

Proof. First we show that ‖L‖χ is a seminorm.Obviously we have ‖L‖χ ∈ [0,∞) for all L ∈ B(X,Y ).Now we show ‖λL‖χ = |λ| ‖L‖χ for all scalars λ and all L ∈ B(X,Y ). Since trivially‖λL‖χ = |λ| ‖L‖χ for λ = 0, we may assume that λ 6= 0. Let ρ = ‖L‖χ and ε > 0 begiven. Then we have χ(L(Q)) ≤ (ρ + ε)χ(Q) and so χ(λL(Q)) = |λ|χ(L(Q)) ≤ |λ|(ρ +ε)χ(Q) for all Q ∈ MX . Since ε > 0 was arbitrary, it follows that χ(λL(Q)) ≤ |λ|χ(Q)for all Q ∈ MX , hence ‖λL‖χ ≤ |λ|ρ = |λ| ‖L‖χ. Also it follows by what we have justshown that ‖L‖χ = ‖λ−1(λL)‖χ ≤ |λ−1| ‖λL‖χ, and so |λ| ‖L‖χ ≤ ‖λL‖χ.Now we prove the triangle inequality.Let ρk = ‖Lk‖χ for k = 1, 2, and ε > 0 be given. Then we have χ(Lk(Q)) ≤ (ρk +ε/2)χ(Q) for all Q ∈MX , and so by Proposition 6.3 (i)

χ ((L1 + L2)(Q)) = χ ((L1(Q) + L2(Q)) ≤ χ(L1(Q)) + χ(L2(Q)) ≤ (ρ1 + ρ2 + ε)χ(Q)

for all Q ∈MX . Since ε > 0 was arbitrary, this implies χ((L1 +L2)(Q)) ≤ (ρ1 +ρ2)χ(Q)for all Q ∈ MX , hence ‖L1 + L2‖χ ≤ ρ1 + ρ2 = ‖L1‖χ + ‖L2‖χ which is the triangleinequality.Thus we have shown that ‖ · ‖χ is a seminorm.

The statement in (6.13) is trivial in view of the remarks at the beginning of thissection.

(i) Let BX and BY denote the closed unit balls in X and Y . If y ∈ L(BX) then there

is x ∈ BX such that y = L(x) and ‖y‖ = ‖L(x)‖ ≤ ‖L‖, hence L(BX) ⊂ BY (0, ‖L‖) ⊂‖L‖ BY , and it follows from (6.11) in Theorem 6.7, Propositions 6.2 (iii) and 6.3 (iii),and Theorem (6.4) that ‖L‖χ = χ(L(BX)) ≤ χ(‖L‖ BY ) = ‖L‖χ(BY ) = ‖L‖.

(ii) Let Q ∈MX be given. Then we have χ((L◦L)(Q)) = χ(L(L(Q))) ≤ ‖L‖χ χ(L(Q))

≤ ‖L‖χ ‖L‖χ χ(Q), and (ii) follows from Definition 6.6. ¤


Now we apply our results to characterise the classes C(`1, `1) and C(c, c). First wecharacterise the class C(`1, `1).

Theorem 6.9. ([M–R, Theorem 2.27, p. 175]) Let L ∈ B(`1, `1), and A denote theinfinite matrix such that L(x) = Ax for all x ∈ `1. Then we have L ∈ C(`1, `1) if andonly if

(6.14) limr→∞

(sup

k

∞∑n=r

|ank|)

= 0.

Proof. By Theorem 3.20 (b), every L ∈ B(X, Y ) can be represented by a matrix A ∈(X, Y ). Writing S = S`1 , we have ‖L‖χ = χ(L(S)) by (6.11) in Theorem 6.7. Forr = 0, 1, . . . , let A(r) be the matrix with the first r rows replaced by 0. Then we obtain‖(I −Pr−1)(L(x))‖1 = ‖A(r)x‖1 hence, by (5.7) in Example 5.5,

supx∈S

‖(I − Pr−1)(L(x))‖1 = ‖A(r)‖(`1,`1) = supk

∞∑n=r

|ank|.

Since obviously ‖I − Pr−1‖ = 1 for all r, and the limit in (6.14) exists, it follows from(6.6) in Theorem 6.5 that χ(L(S)) = limr→∞(supk

∑∞n=r |ank|). Finally it follows from

(6.13) in Theorem 6.8 that L ∈ C(`1, `1) if and only if (6.14) is satisfied. ¤

Remark 6.10. It follows from Theorem 6.9 and Example 5.5 that every L ∈ B(`1, `1) iscompact.

Now we characterise the class C(X, Y ).First we give a representation of continuous linear operators from c to c.

Theorem 6.11. We have L ∈ B(c, c) if and only if there exists a matrix A ∈ (c0, c) anda sequence b ∈ `∞ with

(6.15) limn→∞

(bn +

∞∑

k=0

ank

)= α exists

such that

(6.16) L(x) = b limk→∞

xk + Ax for all x ∈ c;

furthermore, we have

(6.17) ‖L‖ = supn

(|bn|+

∞∑

k=0

|ank|)

.

Proof. First we assume that L ∈ B(c, c).We write Ln = Pn ◦ L (n = 0, 1, . . . ) where Pn is the n–the coordinate with Pn(x) = xn


(x ∈ ω). Since c is a BK space, we have Ln ∈ c∗ for all n, that is by (1.2)

(6.18) Ln(x) = bn limk→∞

xk +∞∑

k=0

ankxk (x ∈ c)

with bn = Ln(e)−∞∑

k=0

Ln(e(k)) and ank = Ln(e(k)) for k = 0, 1, . . . ,

and by (1.3)

(6.19) ‖Ln‖ = |bn|+∞∑

k=0

|ank|.

Now (6.18) yields (6.16). Since L(x0) = Ax0 for all x0, we have A ∈ (c0, c), and so‖A‖ = supn

∑∞k=0|ank| < ∞ by (5.7) in Example 5.5. Also L(e) = b + Ae implies (6.15),

and we obtain ‖b‖∞ ≤ ‖L(e)‖∞ + ‖A‖ < ∞, that is b ∈ `∞. Consequently we haveC = supn(|bn| +

∑∞k=0|ank|) < ∞. Now ‖L(x)‖∞ = supn |bn limk→∞ xk +

∑∞k=0ankxk| ≤

(supn(|bn|+∑∞

k=0|ank|)) ‖x‖∞ implies ‖L‖ ≤ C. We also have |Ln(x)| ≤ ‖L(x)‖∞ ≤ ‖L‖for all x ∈ Bc and all n, and so supn ‖Ln‖ = C ≤ ‖L‖. Thus we have shown (6.17).Conversely we assume that A ∈ (c0, c) and b ∈ `∞ satisfy (6.15). Since A ∈ (c0, c) andb ∈ `∞, we obtain C < ∞ by (5.7) in Example 5.5, and so L ∈ B(c, `∞). Finally let x ∈ cbe given and ξ = limk→∞ xk. Then we have x − ξe ∈ c0, Ln(x) = bnξ +

∑∞k=0ankxk =

(bn +∑∞

k=0ank)ξ + An(x − ξe) for all n, and it follows from (6.15) and A ∈ (c0, c) thatlimn→∞ Ln(x) exists. Since x ∈ c was arbitrary, we have L ∈ B(c, c). ¤

Now we apply Theorems 6.11 and 6.5.

Theorem 6.12. Let L ∈ B(c, c). Using the notations of Theorem 6.11 and writingαk = limn→∞ ank for all k = 0, 1, . . . , we have

(6.20)1

2lim sup

n→∞

(∣∣∣∣∣bn − α +∞∑

k=0

αk

∣∣∣∣∣ +∞∑

k=0

|ank − αk|)≤ ‖L‖χ

≤ lim supn→∞

(∣∣∣∣∣bn − α +∞∑

k=0

αk

∣∣∣∣∣ +∞∑

k=0

|ank − αk|)

Proof. We assume that L ∈ B(c, c).Let x ∈ c be given, ξ = limk→∞ xk and y = L(x). We have y = bξ + Ax where A ∈ (c0, c)and b ∈ `∞ by Theorem 6.11, and also note that the limits αk = limn→∞ ank exist for allk by Example 5.4 (b). We can write

(6.21) yn = bnξ + Anx = ξ

(bn +

∞∑

k=0

ank

)+ An(x− ξe) for all .

Since A ∈ (c0, c) it follows from (5.3) in Example 5.2 that

(6.22) limn→∞

An(x− ξe) =∞∑

k=0

αk(xk − ξ) =∞∑

k=0

αkxk − ξ

∞∑

k=0

αk.


Thus it follows from (6.21), (6.22) and (6.15) that

(6.23) η = limn

yn = ξ

(α−

∞∑

k=0

αk

)+

∞∑

k=0

αkxk.

We are going to apply the Goldenstein–Gohberg–Markus theorem (Theorem 6.5) toestablish the estimate in (6.20).First we note that ‖L‖χ = χ(L(Bc)) by (6.11) in Theorem 6.7. Since every sequencez = (zk)

nk=0 ∈ c has a representation z = ζe +

∑∞k=0(zk − ζ)e(k) with ζ = limk→∞, we

define the projector Pr : c → c by Pr(z) = ζe +∑r

k=0(zk − ζ)e(k), and it follows that thesequence z = (I −P)(z) is given by zk = 0 for 0 ≤ k ≤ r and zk = zk − ζ for k ≥ r + 1.Therefore we have |zk| ≤ |zk|+ |ζ| ≤ 2‖z‖∞ for all k, hence ‖I −Pr‖ ≤ 2. Now let z bethe sequence with zr+1 = (−1) and zk = 1 for k 6= r + 1. Then ζ = 1, ‖z‖∞ = 1 and‖(I −Pr)(z)‖∞ = 2, hence ‖I − Pr‖ = 2. Thus we it follows that

(6.24) limr→∞

‖I − Pr‖ = 2.

Writing fn(x) = ((I −Pr)(L(x)))n, we obtain for n ≥ r + 1 by (6.21) and (6.23)

fn(x) = yn − η = ξbn + An(x)−(

ξ

(α−

∞∑

k=0

αk

)+

∞∑

k=0

αkxk

)

= ξ

(bn − α +

∞∑

k=0

αk

)+

∞∑

k=0

(ank − αk)xk,

and see that fn ∈ c∗ by (1.2), and ‖fn‖ = |bn − α +∑∞

k=0αk|+∑∞

k=0|ank − αk| by (1.3).Thus we have shown that

supx∈Bc

‖(I − Pr)(L(x))‖ = supn≥r+1

‖fn‖ = supn≥r

(∣∣∣∣∣bn − α +∞∑

k=0

αk

∣∣∣∣∣ +∞∑

k=0

|ank − αk|)

,

and (6.20) now follows from (6.24), (6.11) in Theorem 6.7 and (6.6) in Theorem 6.5. ¤

The characterisation of the class C(c, c) is an immediate consequence of Theorem 6.12.

Corollary 6.13. Let L ∈ B(c, c). Then L is compact if and only if

(6.25) limn→∞

(∣∣∣∣∣bn − α +∞∑

k=0

αk

∣∣∣∣∣ +∞∑

k=0

|ank − αk|)

= 0.

In particular, if A ∈ (c, c) then LA is compact if and only if

(6.26) limn→∞

(∣∣∣∣∣∞∑

k=0

αk − a

∣∣∣∣∣ +∞∑

k=0

|ank − αk|)

= 0.

Remark 6.14. It is obvious from the second part of Corollary 6.13 that if A is a regularmatrix then LA cannot be compact. If A is a conservative matrix and LA is compact thenA is conull, that is a−∑∞

k=0 αk = 0.


An operator L ∈ B(c, c) is said to be regular if and only if limn→∞(L(x))n = limk→∞ xk

for all x ∈ c.

Corollary 6.15 (Cohen–Dunford). ([C–D, Corollary 3]) Let L ∈ B(c, c) be regular.Then L is compact if and only if

(6.27) limn→∞

(|bn − 1|+

∞∑

k=0

|ank|)

= 0.

Proof. We show that L ∈ B(c, c) is regular if and only if αk = 0 and α = 1. Then thestatement of the corollary is an immediate consequence of Corollary 6.13.First we assume that L ∈ B(c, c) is regular. By Theorem 6.11 there are a matrix A ∈(c0, c) such that (6.15) holds, and a sequence b ∈ `∞ such that L(x) = b limk→∞ xk +A(x)for all x ∈ c. Thus we have

(6.28) limn→∞

(L(e(k)))n = 0 = limn→∞

ank = 0

and

(6.29) limn→∞

(L(e))n = 1 = limn→∞

(bn +

∞∑

k=0

ank

)= α.

Conversely if L(x) = b limk→∞ xk+A(x) and (6.28) and (6.29) are satisfied then it followsfrom (6.23) that limn→∞(L(x))n = limk→∞ xk(α −

∑∞k=0αk) +

∑∞k=0αkxk = limk xk for

all x ∈ c, and L is regular. ¤

Appendix A. Known Results from Functional Analysis

The following results are well known in functional analysis.

Theorem A.1 (The closed graph lemma). ([Wil1, Theorem 11.1.1, p. 195]) Any con-tinuous map into a Hausdorff space has closed graph.

Theorem A.2 (The closed graph theorem). ([Wil1, Theorem 11.2.2, p. 200]) If X andY are Frechet spaces and f : X → Y is a closed linear map, then f is continuous.

Theorem A.3 (The Banach–Steinhaus theorem). ([Wil1, Corollary 11.2.4, p. 200]) Let(fn) be a pointwise convergent sequence of linear functionals on a Frechet space X. Thenf defined by f(x) = limn→∞ f(x) is continuous.

Theorem A.4 (The Hahn–Banach theorem). ([Wil2, 3.0.4, p. 39]) Let X be a subspaceof a linear topological space Y and f be a linear functional on X which is continuous inthe relative topology of Y . Then f can be extended to a continuous linear functional onY .

Theorem A.5 (The uniform boundedness principle). ([Wil1, Corollary 11.2.3, p. 200])Let (fn) be a pointwise convergent sequence of continuous linear functionals on a Frechetspace. Then (fn) is equicontinuous.


Theorem A.6 (The convergence lemma). ([Wil2, 7.0.3, p.103]) Let (fn) be a sequenceof equicontinuous linear functionals on a linear topological space X. Then the set {x ∈X : limn→∞ fn(x) exists } is a closed linear subspace of X.

Theorem A.7 (The open mapping theorem). ([Wil1, Theorem 11.2.1, p. 199]) Let Xand Y be Frechet spaces, and f : X → Y a closed linear map onto. Then f is open.

References

[AKP] R. R. Akhmerov, M. I. Kamenskiı, A. S. Potapov, B. N. Sadovskiı, Measures of Noncompactnessand Condensing Operators, Operator Theory: Advances and Applications, Vol. 55, Birkhauser,Basel, 1992

[B–G] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure andApplied Mathematics, Vol. 60, Marcel Dekker, New York, 1980

[Boo] J. Boss, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000[C–D] L. W. Cohen, N. Dunford, Transformations on sequence spaces, Duke Math. J. 3(1937), 689–701[Coo] R. C. Cooke, Infinite Matrices and Sequence Spaces, MacMillan and Co. Ltd, London, 1950[Cro] L. Crone, A characterization of matrix mappings on `2, Math. Z. 123 (1971), 315–317[Dar] G. Darbo, Punti uniti in transformazioni a condominio non compatto, Rend. Sem. Math. Univ.

Padova 24 (1955), 84–92[GGM] I. T. Gohberg, L. S. Goldenstein, A. S. Markus, Investigations of some properties with their

q–norms, Ucen. Zap. Kishinevsk. Univ. 29 (1957), 29–36 (Russian)[Har] G. H. Hardy, Divergent Series, Oxford University Press, 1973[Ist] V. Istratesku, On a measure of noncompactness, Bull. Math. Soc. Sci. Math. R. S. Roumanie

(N.S.) 16 (1972), 195–197[Ist1] V. Istratesku, Fixed Point Theory, An Introduction, Reidel, Dordrecht, Boston and London, 1981[J–M] A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability, Filomat 17 (2003),

59–78[K–G] P. K. Kamthan, M. Gupta, Sequence Spaces and Series, Marcel Dekker, New York, 1981[Kur] K. Kuratowski, Sur les espaces complets, Fund. Math 15 (1930), 301–309[Mad] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1971[Mad1] I. J. Maddox, Infinite Matrices of Operators, Lecture Notes in Mathematics 780, Springer

Verlag, Heidelberg–Berlin–New York, 1980[Mal] E. Malkowsky, Matrix Transformations in a New Class of Sequence Spaces that Includes Spaces

of Absolutely and Strongly Summable Sequences, Habilitationsschrift, Giessen, 1988[M–R] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measures

of noncompactness, Zbornik radova, Matematicki institut SANU 9(17) (2000), 143–234[Pey] A. Peyerimhoff, Lectures on Summability, Lecture Notes in Mathematics 107, Springer Verlag,

Heidelberg–Berlin–New York, 1969[Ruc] W. H. Ruckle, Sequence Spaces, Research Notes in Mathematics 49, Pitman, Boston, London,

Melbourne, 1981[S–T] M. Stieglitz, H. Tietz, Matrixtransformationen in Folgenraumen, Math. Z. 154 (1977), 1–16[Toe] O. Toeplitz, Uber allgemeine Mittelbildungen, Prace. Mat. Fiz. 22 (1911), 113–119[TBA] J. M. Ayerbe Toledano, T. Dominguez Benavides, G. Lopez Acedo Measures of Noncompactness

in Metric Fixed Point Theory, Operator Theory, Advances and Applications, Vol. 99, Birkhauser,Basel, Boston, Berlin, 1997

[Wil1] A. Wilansky, Functional Analysis, Blaisdell Publishing Company, New York, Toronto, London,1964

[Wil2] A. Wilansky, Summability through Functional Analysis, North–Holland Mathematical Studies85, Elsevier Science Publishers, Amsterdam, New York, Oxford, 1984

[Zel] K. Zeller, Abschnittskonvergenz in FK–Raumen, Math. Z. 55 (1951), 55–70


[Z–B] K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer Verlag, Heidelberg–Berlin–New York, 1968

Department of Mathematics, University of Giessen, Arndtstrasse 2, D–35392 Giessen,Germany, c/o School of Informatics and Computing, German Jordanian University,Amman, Jordan

E-mail address: [email protected], [email protected]


STRONGLY SINGULAR CALDERON-ZYGMUND OPERATORSAND THEIR COMMUTATORS

YAN LIN AND SHANZHEN LU

Abstract In this paper, the authors obtain two kinds of endpoint estimates forstrongly singular Calderon-Zygmund operators. Moreover, the pointwise estimate forthe sharp maximal function of commutators generated by strongly singular Calderon-Zygmund operators and BMO functions is also established. As its applications, theboundedness of the commutators on Morrey type spaces will be obtained.

1. Introduction

The introduction of strongly singular integral operators is motivated by a class ofmultiplier operators whose symbol is given by ei|ξ|a/|ξ|β away from the origin, where0 < a < 1 and β > 0. Fefferman and Stein [8] have enlarged the multiplier operatorsonto a class of convolution operators. Coifman [6] has also considered a related classof operators for n = 1.

The strongly singular non-convolution operator was introduced by Alvarez andMilman in [3], whose properties are similar to those of Calderon-Zygmund operators,but the kernel is more singular than that of the standard case near the diagonal.Furthermore, following a suggestion of Stein, the authors in [3] showed that the

pseudo-differential operators with symbols in the class S−βα, δ, where 0 < δ ≤ α < 1

and n(1−α)/2 ≤ β < n/2, are included in the strongly singular Calderon-Zygmundoperator defined as follows.

Definition 1.1. Let T : S → S ′ be a bounded linear operator. T is called a stronglysingular Calderon-Zygmund operator if the following conditions are satisfied:

(1) T can be extended into a continuous operator from L2 into itself.

(2) There exists a continuous function K(x, y) away from the diagonal {(x, y) :x = y} such that

|K(x, y)−K(x, z)|+ |K(y, x)−K(z, x)| ≤ C|y − z|δ|x− z|n+ δ

α

,

MR(2000) Subject Classification 42B20, 42B35.Keywords Strongly singular Calderon-Zygmund operator, BMO, LMO, Commutator, Morrey

space.Copyright c© Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan.Received Sept. 2, 2007 Accepted March 4, 2008.

31

if

2|y − z|α ≤ |x− z| for some 0 < δ ≤ 1, 0 < α < 1,

and 〈Tf, g〉 =∫

K(x, y)f(y)g(x)dydx, for f, g ∈ S with disjoint supports.

(3) For some n(1− α)/2 ≤ β < n/2, both T and its conjugate operator T ∗ can beextended into continuous operators from Lq to L2, where 1/q = 1/2 + β/n.

Alvarez and Milman [3, 4] discussed the boundedness of strongly singular Calderon-Zygmund operators on Lebesgue spaces.

Theorem A [3] If T is a strongly singular Calderon-Zygmund operator, then T canbe defined to be a continuous operator from L∞ to BMO.

Theorem B [4] If T is a strongly singular Calderon-Zygmund operator, then T isof weak (L1, L1) type.

Moreover, the authors in [10] obtained that the strongly singular Calderon-Zygmundoperator T is bounded from H1 to L1. Obviously, T is bounded on Lp, 1 < p < ∞,by the interpolation theory.

Now, let us return to the classical singular integral defined by

Tf(x) = limε→0

∫

|x−y|>ε

K(x− y)f(y)dy,

where the kernel K ∈ C(Rn \ {0}) satisfies the following three conditions:

(a)∫

ε<|x|<NK(x)dx = 0, for any 0 < ε < N < ∞;

(b) |K(x)| ≤ C|x|−n, x 6= 0;

(c) |K(x− y)−K(x)| ≤ C|y||x|−n−1, for all |x| > 2|y|.A well-known result in [14] showed that the above operator T is bounded on the

BMO space.

Definition 1.2. LMO is a subspace of BMO, equipped with the semi-norm

[f ]LMO = sup0<r<1

1 + | ln r||Br|

∫

Br

|f(x)− fBr |dx + supr≥1

1

|Br|∫

Br

|f(x)− fBr |dx,

where Br denotes a ball in Rn with radius r.

The authors in [12, 13] also obtained the LMO-boundedness of classical singularintegral operators.

These results mentioned above essentially depend on the cancellation condition ofthe kernel in (a). A natural question is: whether the strongly singular Calderon-Zygmund operator T is bounded on the BMO and LMO spaces if we add a conditionsimilar to (a) to it. In Section 2, we will give an affirmative answer.

32

On the other hand, a pointwise estimate for the sharp maximal function of stronglysingular Calderon-Zygmund operators was obtained in [4]:

(Tf)](x) ≤ CM2(f)(x).

Here and in what follows, for 1 < p < ∞, Mp(f)(x) = M(|f |p)1/p(x), where M standsfor the Hardy-Littlewood maximal function. As a matter of fact, this estimate wasgeneralized in [9] as follows:

(Tf)](x) ≤ CMs(f)(x), for anyn(1− α) + 2β

2β≤ s < ∞,

where α, β are given as in Definition 1.1. Then a weighted norm inequality can

be established immediately. T is bounded on Lpω(Rn) for n(1−α)+2β

2β≤ s < p < ∞

and ω ∈ Ap/s. By the well known result of Alvarez-Bagby-Kurtz-Perez in [2], thecommutator [b, T ] generated by a strongly singular Calderon-Zygmund operator Tand a BMO function b, which is defined by

[b, T ]f(x) = b(x)Tf(x)− T (bf)(x)

for suitable functions f , is also bounded on Lpω(Rn) for n(1−α)+2β

2β≤ s < p < ∞

and ω ∈ Ap/s. In particular, [b, T ] is bounded on Lp(Rn), n(1−α)+2β2β

< p < ∞. A

standard discussion about duality and interpolation yields the boundedness of [b, T ]on Lp(Rn), 1 < p < ∞.

Besides the above method, there is another way to obtain the weighted normestimate of [b, T ]. In Section 3, we will establish a pointwise estimate for the sharpmaximal function of [b, T ] directly. Furthermore, this estimate can be applied toget boundedness properties of [b, T ] on other function spaces. In Section 4, we willstate the boundedness of [b, T ] on Morrey type spaces, which can be regarded asapplications of the result in Section 3.

In what follows, for 1 < p < ∞, p′ is the conjugate index of p, that is, 1/p+1/p′ = 1.χE is the characteristic function of a set E. Ec = Rn \E is the complementary set ofE. C’s will be constants which may vary from line to line. We will always denote byB(x,R) the ball centered at x with radius R > 0, CB(x,R) = B(x,CR) for C > 0,|B(x,R)| the Lebesgue measure of B(x,R) and fB(x, R) = 1

|B(x, R)|∫

B(x, R)f(y)dy.

2.Endpoint estimates

The most useful property of a BMO function is the classical John-Nirenberg in-equality, which shows that functions in BMO are locally exponentially integrable.This implies that for any 1 ≤ q < ∞, the functions in BMO can be described bymeans of the condition:

supB⊂Rn

(1

|B|∫

B

|f(x)− fB|qdx

)1/q

< ∞.

33

Based on the above property, the following estimate can be established, which isvery convenient in applications.

Lemma 2.1. Let f be a function in BMO. Suppose 1 ≤ p < ∞, x ∈ Rn, andr1, r2 > 0. Then

(1

|B(x, r1)|∫

B(x, r1)

|f(y)− fB(x, r2)|pdy

)1/p

≤ C

(1 +

∣∣ln r1

r2

∣∣)‖f‖BMO,

where C > 0 is independent of f , x, r1 and r2.

Proof We only consider the case 0 < r1 ≤ r2. Actually, the similar procedure alsoworks for another case 0 < r2 < r1.

For 0 < r1 ≤ r2, there are k1, k2 ∈ Z such that 2k1−1 < r1 ≤ 2k1 and 2k2−1 < r2 ≤2k2 . Then k1 ≤ k2 and

(k2 − k1 − 1) ln 2 < lnr2

r1

< (k2 − k1 + 1) ln 2.

Thus, we have

(1

|B(x, r1)|∫

B(x, r1)

|f(y)− fB(x, r2)|pdy

)1/p

≤(

1

|B(x, r1)|∫

B(x, r1)

|f(y)− fB(x, 2k1 )|pdy

)1/p

+ |fB(x, 2k1) − fB(x, r2)|

≤(

2n

|B(x, 2k1)|∫

B(x, 2k1 )

|f(y)− fB(x, 2k1 )|pdy

)1/p

+ |fB(x, r2) − fB(x, 2k2 )|

+

k2−1∑

j=k1

|fB(x, 2j+1) − fB(x, 2j)|

≤ C‖f‖BMO +1

|B(x, r2)|∫

B(x, r2)

|f(y)− fB(x, 2k2)|dy

+

k2−1∑

j=k1

1

|B(x, 2j)|∫

B(x, 2j)

|f(y)− fB(x, 2j+1)|dy

≤ C‖f‖BMO +2n

|B(x, 2k2)|∫

B(x, 2k2 )

|f(y)− fB(x, 2k2 )|dy

+

k2−1∑

j=k1

2n

|B(x, 2j+1)|∫

B(x, 2j+1)

|f(y)− fB(x, 2j+1)|dy

≤ ‖f‖BMO

(C + 2n + 2n(k2 − k1)

)34

≤ C

(1 + ln

r2

r1

)‖f‖BMO.

This completes the proof of the lemma. ¤

LMO is essentially a special case of a kind of function spaces introduced by Spannein [12]. For a LMO function, there are some properties similar to those of a BMOfunction. One can refer to [1] for the details.

For 1 ≤ p < ∞, define

[f ]LMOp = sup0<r< 1

2

(1 + | ln r|)(

1

|Br|∫

Br

|f(x)− fBr |pdx

)1/p

.

Lemma 2.2. [1] If f ∈ LMO, then for any 1 ≤ p < ∞, there exists a constantC > 0 depending only on n and p such that

[f ]LMOp ≤ C[f ]LMO.

Lemma 2.3. Let ε > 0 and f ∈ LMO. Then for any ball B = B(x, r) with0 < r < 1

2,

∫

Bc

|f(y)− fB||x− y|n+ε

dy ≤ Cr−ε(1 + | ln r|)−1[f ]LMO,

where C > 0 is independent of f , x and r.

Proof Since r−1 > 2, there exists a k ∈ N+ such that 2k < r−1 ≤ 2k+1. Thenk ∼ | ln r|.

∫

Bc


dy ≤∞∑

j=0

∫

2j+1B\2jB


dy

≤ C

∞∑j=0

(2jr)−ε 1

|2j+1B|∫

2j+1B

|f(y)− fB|dy

≤ C

∞∑j=0

(2jr)−ε

(1

|2j+1B|∫

2j+1B

|f(y)− f2j+1B|dy +

j∑i=0

|f2i+1B − f2iB|)

= C

k−1∑j=0

(2jr)−ε

(1

|2j+1B|∫

2j+1B

|f(y)− f2j+1B|dy +

j∑i=0

|f2i+1B − f2iB|)

+C

∞∑

j=k

(2jr)−ε

(1

|2j+1B|∫

2j+1B

|f(y)− f2j+1B|dy +

j∑i=0

|f2i+1B − f2iB|)

:= I + II.35

Let us estimate II first.

II ≤ C

∞∑

j=k

(2jr)−ε

(1

|2j+1B|∫

2j+1B

|f(y)− f2j+1B|dy

+

j∑i=0

1

|2iB|∫

2iB

|f(y)− f2i+1B|dy

)

≤ C‖f‖BMOr−ε

∞∑

j=k

j 2−εj

≤ Cr−ε[f ]LMO

∞∑

j=k

j2

k + 12−εj

≤ Cr−ε[f ]LMO

∞∑j=1

j2

1 + | ln r|2−εj

= Cr−ε(1 + | ln r|)−1[f ]LMO.

To estimate I, we use the fact that 0 < 2j+1r < 1 for 0 ≤ j ≤ k − 1.

I ≤ C

k−1∑j=0

(2jr)−ε

(1 + | ln 2j+1r|

(1 + | ln 2j+1r|)|2j+1B|∫

2j+1B

|f(y)− f2j+1B|dy

+

j∑i=0

1 + | ln 2i+1r|(1 + | ln 2i+1r|)|2i+1B|

∫

2i+1B

|f(y)− f2i+1B|dy

)

≤ Cr−ε[f ]LMO

k−1∑j=0

2−εj

(1

1 + | ln 2j+1r| +

j∑i=0

1

1 + | ln 2i+1r|)

≤ Cr−ε[f ]LMO

∞∑j=0

j∑i=0

2−εj

1 + | ln 2i+1r|

= Cr−ε[f ]LMO

∞∑i=0

1

1 + | ln r + (i + 1) ln 2|∞∑j=i

2−εj

= Cr−ε[f ]LMO

∞∑i=0

2−εi

1 + | ln r + (i + 1) ln 2|≤ Cr−ε(1 + | ln r|)−1[f ]LMO.

The following basic inequality was applied to get the last inequality above.

1 + |a + b| ≥ (1 + |a|)−1(1 + |b|), for any a, b ∈ R. (2.1)36

Combining the above two estimates, we can obtain the desired result. ¤

Now, let us state the main results in this section.

Theorem 2.1. Let T be a strongly singular Calderon-Zygmund operator and T1 =0. Suppose f ∈ BMO such that Tf(x) exists a.e. in Rn. Then Tf ∈ BMO and

‖Tf‖BMO ≤ C‖f‖BMO,

where C > 0 is independent of f .

Proof Let α, β and δ be given as in Definition 1.1. For any ball B = B(x0, r) ⊂ Rn,there are two cases.

(i) r > 1.

Write

f(x) = f2B + (f(x)− f2B)χ8B(x) + (f(x)− f2B)χ(8B)c(x)

:= f1(x) + f2(x) + f3(x).

It follows from the hypothesis T1 = 0 that Tf1 = 0.

By Holder’s inequality, the L2-boundedness of T and Lemma 2.1, we have

1

|B|∫

B

|Tf2(x)|dx ≤(

1

|B|∫

B

|Tf2(x)|2dx

)1/2

≤ C

(1

|B|∫

Rn

|f2(y)|2dy

)1/2

= C

(1

|8B|∫

8B

|f(y)− f2B|2dy

)1/2

≤ C‖f‖BMO.

Since Tf(x) and Tf2(x) exist a.e. in Rn, there is a point z1 ∈ B such that|Tf3(z1)| < ∞. For any x ∈ B and y ∈ (8B)c, 2|x − z1|α ≤ 2(2r)α < 4r < |y − z1|since r > 1. It follows from (2) of Definition 1.1, Lemma 2.1 and r > 1 that

1

|B|∫

B

|Tf3(x)− Tf3(z1)|dx

≤ 1

|B|∫

B

∫

(8B)c

|K(x, y)−K(z1, y)||f(y)− f2B|dydx

≤ C1

|B|∫

B

∫

(8B)c

|x− z1|δ|y − z1|n+δ/α

|f(y)− f2B|dydx

≤ Crδ

∞∑

k=3

∫

2k+1B\2kB

|f(y)− f2B||y − z1|n+δ/α

dy

37

≤ Crδ

∞∑

k=3

(2kr)−δ/α 1

|2k+1B|∫

2k+1B

|f(y)− f2B|dy

≤ Crδ−δ/α‖f‖BMO

∞∑

k=3

k2−kδ/α

≤ C‖f‖BMO.

The last inequality is due to δ − δ/α < 0.

Thus,

1

|B|∫

B

|Tf(x)− (Tf)B|dx

≤ 2

|B|∫

B

|Tf(x)− Tf3(z1)|dx

≤ 2

|B|∫

B

|Tf2(x)|dx +2

|B|∫

B


≤ C‖f‖BMO.

(ii) 0 < r ≤ 1.

Let B = B(x0, rα). Write

f(x) = f2 eB + (f(x)− f2 eB)χ8 eB(x) + (f(x)− f2 eB)χ(8 eB)c(x)

:= f4(x) + f5(x) + f6(x).


By Holder’s inequality, the (L2, Lq′)-boundedness of T in Definition 1.1, where1/q′ = 1/2− β/n, Lemma 2.1 and 0 < r ≤ 1, we have

1

|B|∫

B

|Tf5(x)|dx ≤(

1

|B|∫

B

|Tf5(x)|q′dx

)1/q′

≤ C|B|−1/q′(∫

8 eB|f(y)− f2 eB|2dy

)1/2

≤ C‖f‖BMO|B|−1/q′|B|1/2

= C‖f‖BMOrn(α/2−1/q′)

≤ C‖f‖BMO.

The last inequality is due to α/2 − 1/q′ ≥ 0 which follows from β ≥ n(1 − α)/2 inDefinition 1.1.

Since Tf(x) and Tf5(x) exist a.e. in Rn, there is a point z2 ∈ B such that

|Tf6(z2)| < ∞. For any x ∈ B and y ∈ (8B)c, 2|x− z2|α ≤ 2(2r)α < 4rα < |y − z2|38

since 0 < r ≤ 1. It follows from (2) of Definition 1.1 and Lemma 2.1 that

1

|B|∫

B


≤ 1

|B|∫

B

∫

(8 eB)c

|K(x, y)−K(z2, y)||f(y)− f2 eB|dydx

≤ C1

|B|∫

B

∫

(8 eB)c

|x− z2|δ|y − z2|n+δ/α

|f(y)− f2 eB|dydx

≤ Crδ

∞∑

k=3

∫

2k+1 eB\2k eB

|f(y)− f2 eB||y − z2|n+δ/α

dy

≤ Crδ

∞∑

k=3

(2krα)−δ/α 1

|2k+1B|

∫

2k+1 eB|f(y)− f2 eB|dy

≤ C‖f‖BMO

∞∑

k=3

k2−kδ/α

= C‖f‖BMO.

Thus,

1

|B|∫

B

|Tf(x)− (Tf)B|dx

≤ 2

|B|∫

B

|Tf(x)− Tf6(z2)|dx

≤ 2

|B|∫

B

|Tf5(x)|dx +2

|B|∫

B


≤ C‖f‖BMO.

Therefore, in both cases, we have

1

|B|∫

B

|Tf(x)− (Tf)B|dx ≤ C‖f‖BMO,

which completes the proof of the theorem. ¤

Remark 2.1. If we assume that T ∗1 = 0, then by a discussion similar to that inTheorem 2.1, it follows that

‖T ∗f‖BMO ≤ C‖f‖BMO

for f ∈ BMO such that T ∗f(x) exists a.e. in Rn.39

Given f ∈ H1, for any g ∈ VMO with compact support, the duality relation(H1)′ = BMO and T ∗1 = 0 imply that

|〈Tf, g〉| = |〈f, T ∗g〉| ≤ ‖T ∗g‖BMO‖f‖H1 ≤ C‖g‖BMO‖f‖H1 .

Because the set of VMO functions with compact support is dense in VMO, we canget that Tf ∈ (VMO)′ = H1. Moreover

‖Tf‖H1 ≤ C‖f‖H1 .

Actually, this conclusion has been obtained by Alvarez and Milman in [3].

From a contrasting point of view, the BMO-boundedness in Theorem 2.1 can bealso formulated by a duality discussion based on the (H1, H1)-boundedness in [3].As a matter of fact, we give a straightforward proof for it in this paper.

On the other hand, the boundedness of strongly singular Calderon-Zygmund op-erators on the LMO space can be established as follows.

Theorem 2.2. Let T be a strongly singular Calderon-Zygmund operator and T1 =0. Suppose f ∈ LMO such that Tf(x) exists a.e. in Rn. Then Tf ∈ LMO and

[Tf ]LMO ≤ C[f ]LMO,

where C > 0 is independent of f .

Proof Let α, β and δ be given as in Definition 1.1. For any ball B = B(x0, r) ⊂ Rn

with r ≥ 1, by the BMO-boundedness of T in Theorem 2.1, we have

1

|B|∫

B

|Tf(x)− (Tf)B|dx ≤ ‖Tf‖BMO ≤ C‖f‖BMO ≤ C[f ]LMO.

It suffices to prove that, for any ball B = B(x0, r) ⊂ Rn with 0 < r < 1, thefollowing inequality holds.

1 + | ln r||B|

∫

B

|Tf(x)− (Tf)B|dx ≤ C[f ]LMO.

We consider two cases respectively.

(i) 16−1/α ≤ r < 1.

The BMO-boundedness of T also implies that

1 + | ln r||B|

∫

B

|Tf(x)− (Tf)B|dx

=1 + ln 1

r

|B|∫

B

|Tf(x)− (Tf)B|dx

≤ C1

|B|∫

B

|Tf(x)− (Tf)B|dx

≤ C‖Tf‖BMO ≤ C‖f‖BMO ≤ C[f ]LMO.40

(ii) 0 < r < 16−1/α.

Let B = B(x0, rα). Write

f(x) = f8 eB + (f(x)− f8 eB)χ8 eB(x) + (f(x)− f8 eB)χ(8 eB)c(x)

:= f1(x) + f2(x) + f3(x).


Notice that 0 < 8rα < 1/2. By Holder’s inequality, the (L2, Lq′)-boundedness ofT , Lemma 2.2 and (2.1), we have

1

|B|∫

B

|Tf2(x)|dx ≤(

1

|B|∫

B

|Tf2(x)|q′dx

)1/q′

≤ C|B|−1/q′(∫

8 eB|f(y)− f8 eB|2dy

)1/2

≤ C[f ]LMO2|B|−1/q′|B|1/2(1 + | ln 8rα|)−1

≤ C[f ]LMOrn(α/2−1/q′)(1 + | ln 8 + α ln r|)−1

≤ C[f ]LMO(1 + ln 8)(1 + α| ln r|)−1

≤ C[f ]LMO(1 + | ln r|)−1.

Since Tf(x) and Tf2(x) exist a.e. in Rn, there is a point x∗ ∈ B such that

|Tf3(x∗)| < ∞. For any x ∈ B and y ∈ (8B)c, 2|x− x∗|α ≤ 2(2r)α < 4rα < |y − x∗|

since 0 < r < 1. It follows from (2) of Definition 1.1, Lemma 2.3 and (2.1) that

1

|B|∫

B

|Tf3(x)− Tf3(x∗)|dx

≤ 1

|B|∫

B

∫

(8 eB)c

|K(x, y)−K(x∗, y)||f(y)− f8 eB|dydx

≤ C1

|B|∫

B

∫

(8 eB)c

|x− x∗|δ|y − x∗|n+δ/α

|f(y)− f8 eB|dydx

≤ Crδ

∫

(8 eB)c

|f(y)− f8 eB||y − x0|n+δ/α

dy

≤ Crδ(8rα)−δ/α(1 + | ln 8rα|)−1[f ]LMO

≤ C[f ]LMO(1 + | ln r|)−1.

Thus,

1 + | ln r||B|

∫

B

|Tf(x)− (Tf)B|dx

41

≤ 21 + | ln r||B|

∫

B

|Tf(x)− Tf3(x∗)|dx

≤ 21 + | ln r||B|

∫

B

|Tf2(x)|dx + 21 + | ln r||B|

∫

B

|Tf3(x)− Tf3(x∗)|dx

≤ C[f ]LMO.

This gives the desired result. ¤

Remark 2.2. It should be pointed out that there is a counterpart of the above resultfor T ∗ under the hypothesis T ∗1 = 0, but we omit the details for their similarity.

3.A pointwise estimate for the sharp maximal function

The definition and properties of BMO functions lead us naturally to study thesharp maximal function f ], associated to any locally integrable function f . It isdefined by

f ](x) = supB3x

1

|B|∫

B

|f(y)− fB|dy

∼ supB3x

infa∈C

1

|B|∫

B

|f(y)− a|dx,

where the supremum is taken over all balls B containing x. In fact, the abovedefinition is equivalent to the one by taking the supremum over all balls B centeredat x.

A function f is in the BMO exactly when f ] is a bounded function. This obser-vation illustrates that sometimes significant aspects of f are most directly expressedin terms of f ].

In this section, we will state a pointwise estimate for the sharp maximal function ofcommutators generated by strongly singular Calderon-Zygmund operators and BMOfunctions. First, the following elementary inequality is necessary.

Lemma 3.1. Given ε > 0, there is

ln x ≤ 1

εxε, for all x ≥ 1.

Let ϕ(x) = ln x− 1εxε, x ≥ 1. The above result comes from the monotone property

of the function ϕ.

Besides the (Lp, Lp)-boundedness, 1 < p < ∞, the strongly singular Calderon-Zygmund operator T still has other boundedness properties on Lebesgue spaces. Byinterpolating between (L2, Lq′) and (L∞, BMO), where q is given as in Definition 1.1

and 1/q + 1/q′ = 1, T is bounded from Lu to Lv, 2 ≤ u < ∞ and v = uq′2

. It is easy

to see that 0 < uv≤ α. Then we interpolate between (L2, Lq′) and weak (L1, L1)

42

to obtain the boundedness of T from Lu to Lv, 1 < u ≤ 2 and v = uq′2q′−uq′+2u−2

.

In this situation, 0 < uv≤ α if and only if n(1−α)+2β

2β≤ u ≤ 2. In a word, T is

bounded from Lu to Lv, n(1−α)+2β2β

≤ u < ∞ and 0 < uv≤ α. In particular, if we

restrict n(1−α)2

< β < n2

in (3) of Definition 1.1, then T is bounded from Lu to Lv,n(1−α)+2β

2β< u < ∞ and 0 < u

v< α.

Theorem 3.1. Let T be a strongly singular Calderon-Zygmund operator, α, β, δ be

given as in Definition 1.1 and n(1−α)2

< β < n2. If b ∈ BMO, then for any s satisfying

n(1−α)+2β2β

< s < ∞, there exists a constant C > 0 such that for all smooth functions

f with compact support,

([b, T ]f)](x) ≤ C‖b‖BMO(Ms(Tf)(x) + Ms(f)(x)).

Proof For any ball B = B(x0, r) ⊂ Rn, there are two cases.

(i) r > 1.

Write

[b, T ]f(x) = [b− b2B, T ]f(x)

= (b− b2B)Tf(x)− T ((b− b2B)fχ2B)(x)− T ((b− b2B)fχ(2B)c)(x).

Then

1

|B|∫

B

|[b, T ]f(x)− T ((b2B − b)fχ(2B)c)(x0)|dx

≤ 1

|B|∫

B

|b(x)− b2B||Tf(x)|dx +1

|B|∫

B

|T ((b− b2B)fχ2B)(x)|dx

+1

|B|∫

B

|T ((b− b2B)fχ(2B)c)(x)− T ((b− b2B)fχ(2B)c)(x0)|dx

:= I1 + I2 + I3.

For I1, Holder’s inequality yields that

I1 ≤(

1

|B|∫

B

|b(x)− b2B|s′dx

)1/s′(1

|B|∫

B

|Tf(x)|sdx

)1/s

≤ C‖b‖BMOMs(Tf)(x0).

Since n(1−α)+2β2β

< s < ∞, there exists an s0 such that n(1−α)+2β2β

< s0 < s < ∞.

Denote by 1/s1 = 1/s0 − 1/s. To estimate I2, we use Holder’s inequality and the(Ls0 , Ls0)-boundedness of T .

I2 ≤(

1

|B|∫

B

|T ((b− b2B)fχ2B)(x)|s0dx

)1/s0

43

≤ C

(1

|B|∫

2B

|b(y)− b2B|s0|f(y)|s0dy

)1/s0

≤ C

(1

|2B|∫

2B

|b(y)− b2B|s1dy

)1/s1(

1

|2B|∫

2B

|f(y)|sdy

)1/s

≤ C‖b‖BMOMs(f)(x0).

For any x ∈ B and y ∈ (2B)c, 2|x − x0|α ≤ 2rα < 2r ≤ |y − x0| since r > 1. Itfollows from (2) of Definition 1.1 and Lemma 2.1 that

I3 ≤ 1

|B|∫

B

∫

(2B)c

|K(x, y)−K(x0, y)||b(y)− b2B||f(y)|dydx

≤ C1

|B|∫

B

∫

(2B)c

|x− x0|δ|y − x0|n+δ/α

|b(y)− b2B||f(y)|dydx

≤ Crδ

∞∑

k=1

∫

2k+1B\2kB

1

|y − x0|n+δ/α|b(y)− b2B||f(y)|dy

≤ Crδ

∞∑

k=1

(2kr)−δ/α 1

|2k+1B|∫

2k+1B

|b(y)− b2B||f(y)|dy

≤ Crδ−δ/α

∞∑

k=1

2−kδ/α

(1

|2k+1B|∫

2k+1B

|b(y)− b2B|s′dy

)1/s′

×(

1

|2k+1B|∫

2k+1B

|f(y)|sdy

)1/s

≤ C‖b‖BMOMs(f)(x0)rδ−δ/α

∞∑

k=1

k2−kδ/α

≤ C‖b‖BMOMs(f)(x0).

(ii) 0 < r ≤ 1.

For the index s0 which we chose above, there exists an l0 such that T is boundedfrom Ls0 to Ll0 and 0 < s0

l0< α. Then we can take a θ satisfying 0 < s0

l0< θ < α.

Let B = B(x0, rθ).

Write

[b, T ]f(x) = [b− b2B, T ]f(x)

= (b− b2B)Tf(x)− T ((b− b2B)fχ2 eB)(x)− T ((b− b2B)fχ(2 eB)c)(x).

Then

1

|B|∫

B

|[b, T ]f(x)− T ((b2B − b)fχ(2 eB)c)(x0)|dx

44

≤ 1

|B|∫

B

|b(x)− b2B||Tf(x)|dx +1

|B|∫

B

|T ((b− b2B)fχ2 eB)(x)|dx

+1

|B|∫

B

|T ((b− b2B)fχ(2 eB)c)(x)− T ((b− b2B)fχ(2 eB)c)(x0)|dx

:= II1 + II2 + II3.

The estimate of II1 is the same as that of I1.

II1 ≤ C‖b‖BMOMs(Tf)(x0).

The inequality 0 < s0

l0< θ implies that ε1 := n( θ

s0− 1

l0) > 0. By Holder’s inequality,

the (Ls0 , Ll0)-boundedness of T , Lemma 2.1 and Lemma 3.1, we have

II2 ≤(

1

|B|∫

B

|T ((b− b2B)fχ2 eB)(x)|l0dx

)1/l0

≤ C|B|−1/l0

(∫

2 eB|b(y)− b2B|s0|f(y)|s0dy

)1/s0

≤ C|B|−1/l0|B|1/s0

(1

|2B|

∫

2 eB|b(y)− b2B|s1dy

)1/s1(

1

|2B|

∫

2 eB|f(y)|sdy

)1/s

≤ C‖b‖BMOMs(f)(x0)|B|−1/l0|B|1/s0

(1 + (1− θ) ln

1

r

)

≤ C‖b‖BMOMs(f)(x0)|B|−1/l0|B|1/s0

(1 +

1

ε1

r−ε1

)

≤ C‖b‖BMOMs(f)(x0)rn( θ

s0− 1

l0)−ε1

= C‖b‖BMOMs(f)(x0).

The fact θ < α implies that ε2 := δα(α − θ) > 0. For any x ∈ B and y ∈ (2B)c,

we have 2|x − x0|α ≤ 2rα ≤ 2rθ ≤ |y − x0| since 0 < r ≤ 1. It follows from (2) ofDefinition 1.1, Lemma 2.1 and Lemma 3.1 that

II3 ≤ 1

|B|∫

B

∫

(2 eB)c

|K(x, y)−K(x0, y)||b(y)− b2B||f(y)|dydx

≤ C1

|B|∫

B

∫

(2 eB)c

|x− x0|δ|y − x0|n+δ/α

|b(y)− b2B||f(y)|dydx

≤ Crδ

∞∑

k=1

∫

2k+1 eB\2k eB

1

|y − x0|n+δ/α|b(y)− b2B||f(y)|dy

≤ Crδ

∞∑

k=1

(2krθ)−δ/α 1

|2k+1B|

∫

2k+1 eB|b(y)− b2B||f(y)|dy

45

≤ Crδ−θδ/α

∞∑

k=1

2−kδ/α

(1

|2k+1B|

∫

2k+1 eB|b(y)− b2B|s′dy

)1/s′

×(

1

|2k+1B|

∫

2k+1 eB|f(y)|sdy

)1/s

≤ C‖b‖BMOMs(f)(x0)rδα

(α−θ)

∞∑

k=1

2−kδ/α

(1 + k + (1− θ) ln

1

r

)


(α−θ)

∞∑

k=1

2−kδ/α

(1 + k +

1

ε2

r−ε2

)


(α−θ)−ε2

∞∑

k=1

k2−kδ/α

= C‖b‖BMOMs(f)(x0).

Thus,

([b, T ]f)](x0) ∼ supB(x0, r)⊂Rn

infa∈C

1

|B|∫

B

|[b, T ]f(x)− a|dx

≤ C‖b‖BMO(Ms(Tf)(x0) + Ms(f)(x0)),

which completes the proof of the theorem. ¤

4.Applications

The estimate for the sharp maximal function of [b, T ] can be applied to obtainnot only the weighted norm estimate of the commutator, but also the boundednessproperties of it on Morrey type spaces.

Morrey spaces have been of great value through the years in studying the localbehavior of solutions to second elliptic partial differential equations.

Definition 4.1. A function f ∈ Lploc(Rn) is said to belong to the classical Morrey

space M qp (Rn), 1 ≤ p ≤ q < ∞, if

‖f‖Mqp (Rn) = sup

B⊂Rn

|B| 1q− 1p

(∫

B

|f(x)|pdx

) 1p

< ∞.

Remark 4.1. It can be seen from the special case Mpp (Rn) = Lp(Rn) with 1 ≤ p <

∞ that Morrey spaces are the generalization of Lebesgue spaces.

Definition 4.2. For a general positive function ϕ on Rn × R+, the generalizedMorrey space Lp, ϕ with 1 ≤ p < ∞ is defined as follows.

Lp, ϕ = {f ∈ Lploc(R

n), ‖f‖Lp, ϕ < +∞},46

where

‖f‖Lp, ϕ = supx∈Rn, r>0

(1

ϕ(x, r)

∫

B(x,r)

|f(y)|pdy

)1/p

.

Remark 4.2. For the case ϕ(x, r) = rn(1−p/q), we have Lp, ϕ = M qp (Rn), 1 ≤ p ≤

q < ∞. Thus, generalized Morrey spaces are the generalization of classical Morreyspaces.

Lemma 4.1. [11] Let ϕ be a positive function on Rn × R+ and there exists a C0

satisfying 0 < C0 < 2n such that

ϕ(x, 2r) ≤ C0ϕ(x, r) for all x ∈ Rn, r > 0. (4.1)

If 1 < p < ∞, then

‖Mf‖Lp, ϕ ≤ C‖f‖Lp, ϕ and ‖Mf‖Lp, ϕ ≤ C‖f ]‖Lp, ϕ ,

where C is independent of f .

Remark 4.3. As a matter of fact, the conditions of ϕ are stronger in [11] thanhere. However, just for the result of Lemma 4.1, the hypothesis here is sufficient.

The boundedness of classical Calderon-Zygmund operators on Morrey spaces wasestablished by Chiarenza and Frasca in [5]. More generally, the authors in [7] ob-tained that a sublinear operator T is bounded on Morrey spaces if T is bounded onLp(Rn) and satisfies the following size condition:

|Tf(x)| ≤ C

∫

Rn

|f(y)||x− y|n dy,

for any f ∈ L1(Rn) with compact support and x /∈ suppf .

For the case when T is a strongly singular Calderon-Zygmund operator, the cor-responding conclusion has been obtained in [9].

Lemma 4.2. [9] Let T be a strongly singular Calderon-Zygmund operator, and α,β, δ be given as in Definition 1.1. Let ϕ be a positive function on Rn×R+ such that

(4.1) holds. If n(1−α)+2β2β

< p < ∞, then T is bounded on Lp, ϕ.

Now, let us proceed with the boundedness of commutators generated by stronglysingular Calderon-Zygmund operators and BMO functions on Morrey spaces.

Theorem 4.1. Let T be a strongly singular Calderon-Zygmund operator, α, β, δ

be given as in Definition 1.1 and n(1−α)2

< β < n2. Let ϕ be a positive function on

Rn × R+ such that (4.1) holds. If b ∈ BMO, then [b, T ] is bounded on Lp, ϕ, wheren(1−α)+2β

2β< p < ∞.

47

Proof Noticing that n(1−α)+2β2β

< p < ∞, there exists an s such that n(1−α)+2β2β

<

s < p < ∞. By Lemma 4.1, Theorem 3.1 and Lemma 4.2, we have

‖[b, T ]f‖Lp, ϕ ≤ ‖M([b, T ]f)‖Lp, ϕ ≤ C‖([b, T ]f)]‖Lp, ϕ

≤ C‖b‖BMO

(‖Ms(Tf)‖Lp, ϕ + ‖Ms(f)‖Lp, ϕ

)

= C‖b‖BMO

(‖M(|Tf |s)‖1/s

Lp/s, ϕ + ‖M(|f |s)‖1/s

Lp/s, ϕ

)

≤ C‖b‖BMO

(‖|Tf |s‖1/s

Lp/s, ϕ + ‖|f |s‖1/s

Lp/s, ϕ

)

= C‖b‖BMO

(‖Tf‖Lp, ϕ + ‖f‖Lp, ϕ

)

≤ C‖b‖BMO‖f‖Lp, ϕ .

This completes the proof of the theorem. ¤

In particular, if we take ϕ(x, r) = rn(1−p/q), 1 ≤ p ≤ q < ∞, then Remark 4.2implies the following conclusion.

Corollary 4.1. Let T be a strongly singular Calderon-Zygmund operator, α, β, δ

be given as in Definition 1.1 and n(1−α)2

< β < n2. If b ∈ BMO, then [b, T ] is bounded

on M qp (Rn), where n(1−α)+2β

2β< p ≤ q < ∞.

Acknowledgement This research was supported by the National Natural ScienceFoundation of China (10571014) and the Doctoral Programme Foundation of Institu-tion of Higher Education of China (20040027001). The authors thank to the refereefor his useful comments.

References

[1] P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl.161(1992), 231-269.

[2] J. Alvarez, R. J. Bagby, D. S. Kurtz, C. Perez, Weighted estimates for commutators of linearoperators, Studia Math. 104(1993), 195-209.

[3] J. Alvarez, M. Milman, Hp continuity properties of Calderon-Zygmund-type operators, J.Math. Anal. Appl. 118(1986), 63-79.

[4] J. Alvarez, M. Milman, Vector valued inequalities for strongly singular Calderon-Zygmundoperators, Rev. Mat. Iberoamericana 2(1986), 405-426.

[5] F. Chiarenza, M. Frasca, Morrey space and Hardy-Littlewood maximal function, Rend. Mat.7(1987), 273-279.

[6] R. Coifman, A real variable characterization of Hp, Studia Math. 51(1974), 269-274.[7] D. Fan, S. Z. Lu, D. C. Yang, Regularity in Morrey spaces of strong solutions to nondivergence

elliptic equations with VMO coefficients, Georgian Math. 5(1998), 425-440.[8] C. Fefferman, E. M. Stein, Hp spaces of several variables, Acta Math. 129(1972), 137-193.[9] Y. Lin, Strongly singular Calderon-Zygmund operator and commutator on Morrey type spaces,

Acta Math. Sinica (English Ser.) (23)2007, 2097-2110.[10] Y. Lin, S. Z. Lu, Toeplitz operators related to strongly singular Calderon-Zygmund operators,

Science in China (Ser. A) 49(2006), 1048-1064.48

[11] L. Z. Liu, Interior estimates in Morrey spaces for solutions of elliptic equations and weightedboundedness for commutators of singular integral operators, Acta Math. Scientia 25(2005),89-94.

[12] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. ScuolaNorm. Sup. Pisa. 19(1965), 593-608.

[13] Y. Z. Sun, W. Y. Su, An endpoint estimate for the commutator of singular integrals, ActaMath. Sinica 21(2005), 1249-1258.

[14] E.M. Stein(1993), Harmonic Analysis: Real-Variable Methods, Orthogonality, and OscillatoryIntegrals, Princeton Univ. Press, Princeton, New Jersey.

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R.China

E-mail address: [email protected], [email protected] (Corresponding author)

49


MATRIX TRANSFORMATIONS BETWEEN SETS OFTHE FORM Wξ AND OPERATOR GENERATORS OF

ANALYTIC SEMIGROUPS

BRUNO DE MALAFOSSE AND EBERHARD MALKOWSKY

Abstract. In this paper we establish a relation between the no-tion of operators of analytic semigroups and matrix transforma-tions from a set of sequences into w∞. We get extensions of someresults given by Labbas and de Malafosse concerning applicationsof the sum of operators in the nondifferential case.

1. Introduction

In this paper we consider spaces that generalize the well-known setsw0 and w∞ introduced and studied by Maddox [5]. Recall that w0 andw∞ are the sets of sequences that are strongly summable to zero andbounded by the Cesaro method of order 1. In [14], Malkowsky andRakocevic gave the characterizations of matrix maps between w0, w,or w∞ and wp

∞ and between w0, w, or w∞ and `1.More recently it was shown by de Malafosse and Malkowsky in [10]

that if λ is an exponentially bounded sequence then (w∞(λ), w∞(λ))is a Banach algebra. Here we give some properties of operators onthe sets Wτ = Dτw∞ and apply these results to particular matrixtransformations between Wτ and w∞. In this way we are led to explictlyrepresent two unbounded operators that are given by infinite matricesand are operator generators of an analytic semigroup (OGASG). Recallthat this notion is a part of the theory of the sum of operators whichwas studied by many authors such as Da Prato and Grisvard [1], R.Labbas and B. Terreni [3]. In Labbas and de Malafosse [4] and deMalafosse [7] there are some applications of the sum of operators inthe theory of summability in the noncommutative case. In this paperwe extend some results given in [4, 7] using the same infinite matrices

1991 Mathematics Subject Classification. Primary: 35A35, Secondary 40C05,46A45.

Key words and phrases. Strongly bounded sequences, BK space, generators ofanalytic semigroup, Banach algebra, bounded operator.

Copyright c© Deanship of Research and Graduate Studies, Yarmouk University,Irbid, Jordan.

Received Sept. 1, 2007 Accepted March 4, 2008.51

52 BRUNO DE MALAFOSSE AND EBERHARD MALKOWSKY

A and B defined here in sets Wξ ⊂ w∞. The relative boundedness withrespect to A or B is not satisfied, so we are not within the frameworkof the classical perturbation theory given by Kato [2].

In this paper we establish a relation between results in summabilityand the basic notions used in the theory of the sum of operators.

2. Preliminaries and well known results

For a given infinite matrix M = (anm)n,m≥1 we define the operatorsMn for every integer n ≥ 1 by Mn(X) =

∑∞m=1 anmxm, where X =

(xn)n≥1 and the series are convergent. So we are led to the study of theinfinite linear system Mn(X) = yn with n = 1, 2, ... where Y = (yn)n≥1

is a one-column matrix and X is the unknown, see [6, 9]. The equationsMn(X) = yn for n = 1, 2, ... can be written in the form MX = Y , whereMX = (Mn(X))n≥1. We write s for the set of all complex sequencesand `∞, c0 for the set of all bounded and null sequences. Recall that`∞ and c0 are Banach spaces with norm ‖X‖`∞ = supn≥1(|xn|).

For subsets E and F of s we denote by (E, F ) the set of all matricesthat map E to F . For any subset E of s, we write ME for the set ofall Y ∈ s such that Y = MX for some X ∈ E. If F is a subset of s,we will denote

(1) F (M) = FM = {X ∈ s : Y = MX ∈ F}.A Banach space E of complex sequences with the norm ‖ · ‖E is aBK space if each projection Pn : X 7→ PnX = xn is continuous,(cf. [15]). A BK space E ⊂ s is said to have AK if every sequenceX = (xn)n≥1 ∈ E has a unique representation X =

∑∞n=1 xne(n) where

e(n) is the sequence with 1 in the n-th position and 0 otherwise. Theset B(E) of all bounded linear operators L mapping E to E normedby

‖L‖∗B(E) = supX 6=0

‖L (X)‖E

‖X‖E

is a Banach algebra and it is well known that if E is a BK space withAK, then B(E) = (E, E).

Throughout we write U+ for the set of all sequences (un)n≥1 withun > 0 for all n, and e = (1, ..., 1, ...). For λ = (λn)n≥1 ∈ U+ we definethe operator C(λ) = (cnm)n,m≥1 by

cnm =

1

λn

if m ≤ n,

0 otherwise.(n = 1, 2, . . . ).

MATRIX TRANSFORMATIONS AND OPERATOR GENERATORS 53

It can be proved that the matrix ∆(λ) = (c′nm)n,m≥1 with

c′nm =

λn if m = n,

−λn−1 if m = n− 1 and n ≥ 2,

0 otherwise

is the inverse of C(λ), see [8]. In the following we use the spaces ofstrongly bounded and summable sequences defined by

w∞(λ) = {X = (xn)n≥1 ∈ s : C(λ)|X| ∈ `∞ },w0(λ) = {X ∈ s : C(λ)|X| ∈ c0 }

and

w(λ) = {X ∈ s : X − le ∈ w0(λ) for some l ∈ C}.These spaces were studied by Malkowsky with the concept of exponen-tially bounded sequences, see [12]. Recall that Maddox [5] defined andstudied the special case λn = n for all n of these sets, and denotedthem by w∞, w0 and w.

3. The set Wτ and matrix transformations between setsof the form Wξ

In this section we state some results on Wτ = Dτw∞ and deal withthe triangles ∆ρ and ∆T

ρ that map Wτ to itself.

3.1. Some properties of the set Wτ . For a given sequence τ =(τn)n≥1 ∈ U+, we define the infinite diagonal matrix Dτ = (τnδnm)n,m≥1.For any subset E of s, DτE is the set of sequences with (xn/τn)n≥1 ∈ E.We put Wτ = Dτw∞, W 0

τ = Dτw0 for τ ∈ U+. So We = w∞ and

W 0e = w0. It is well known [5] that w∞ and w0 are BK spaces with the

norm

‖X‖w∞ = supn

(1

n

n∑m=1

|xm|)

and w0 has AK. It was shown in [10] that the class (w∞, w∞) is aBanach algebra normed by

(2) ‖M‖∗(w∞,w∞) = supX 6=0

(‖MX‖w∞

‖X‖w∞

).

To study operator generators of analytic semigroups (OGASG) map-ping into w∞ we need the following results.


Proposition 3.1. Let τ , ν ∈ U+. Theni) The sets Wτ and W 0

τ are BK spaces normed by

(3) ‖X‖Wτ = supn

(1

n

n∑m=1

|xm|τm

)

and W 0τ has AK.

ii) We have τ/ν ∈ `∞ if and only Wτ ⊂ Wν.iii) We have Wτ = Wν if and only if there are C1, C2 > 0 with

(4) C1 ≤ τn

νn

≤ C2 for all n.

Proof. i) Since w∞ and w0 are BK spaces with the norm ‖ · ‖w∞ by [14,Theorem 3.3, pp. 179], the sets Wτ = w∞(D1/τ ) and W 0

τ = w0(D1/τ )are BK spaces normed by ‖X‖Wτ = ‖D1/τX‖w∞ and W 0

τ has AK.ii) The necessity is a direct consequence of the inequality

‖X‖Wν ≤∥∥∥τ

ν

∥∥∥`∞‖X‖Wτ for all X ∈ Wτ .

Conversely put λn = n for all n. The inclusion Wτ ⊂ Wν means thaty = C(λ)D1/τ |X| ∈ `∞ implies C(λ)D1/ν |X| ∈ `∞ for all X ∈ s. Since(C(λ)D1/τ )

−1 = Dτ∆(λ), we have Wτ ⊂ Wν if and only if

Y ∈ `∞ implies C(λ)D1/νDτ∆(λ)Y ∈ `∞ for all Y,

that is

(5) C(λ)Dτ/ν∆(λ) ∈ (`∞, `∞).

An elementary calculation gives

[C(λ)Dτ/ν∆(λ)

]nm

=

(τm

νm

− τm+1

νm+1

)m

nfor m ≤ n− 1,

τn

νn

for m = n,

0 otherwise.

Then using the characterization of (`∞, `∞) and condition (5), we ob-tain

τn

νn

≤ supn≥2

(n−1∑m=1

∣∣∣∣τm

νm

− τm+1

νm+1

∣∣∣∣m

n+

τn

νn

)< ∞ for all n

and we conclude τ/ν ∈ `∞.iii) Wτ = Wν is equivalent to τ/ν, ν/τ ∈ `∞ that is (4). ¤


3.2. On the operators ∆+ρ and ∆−

ρ considered as maps in Wτ .For given a given sequence ρ = (ρn)n≥1 we consider the operator ∆+

ρ

defined by [∆+

ρ X]n

= xn − ρnxn+1 for all n ≥ 1.

Then we get, putting (∆+ρ )T = ∆−

ρ ,[∆−

ρ X]n

= xn − ρn−1xn−1 for all n ≥ 1

with the convention x0 = 0.To state the next Lemma we write for all τ ∈ U+ and all integers k

ρ+(τ) = (ρnτn+1/τn)n≥1, ρ−(τ) = (ρnτn−1/τn)n≥2,

θ+k (τ) = (1 +

1

k) sup

n≥k(|ρ+

n (τ)|) and θ−k (τ) = (1 +1

k) sup

n≥k+1

(|ρ−n (τ)|) .

We also use the infinite matrices Σ+(N)ρ and Σ

−(N)ρ defined by

Σ+(N)ρ =

[∆

+(N)ρ

]−1

0

10 .

and Σ−(N)

ρ =

[∆−(N)ρ

]−1

0

10 .

where ∆+(N)ρ and ∆

−(N)ρ are the finite matrices whose elements are those

of the N first rows and columns of ∆+ρ and ∆−

ρ . Now we state the nextlemma.

Lemma 3.2. Let ρ, τ ∈ U+.

i) a) For every N ≥ 1 we have ‖I −∆+ρ Σ

+(N)ρ ‖∗(Wτ ,Wτ ) ≤ θ+

N(τ);

b) if limn→∞|ρ+n (τ)| = 0 then

limn→∞

∥∥I −∆+ρ Σ+(n)

ρ

∥∥∗(Wτ ,Wτ )

= 0.

ii) a) For every N ≥ 1 we have ‖I −∆−ρ Σ

−(N)ρ ‖∗(Wτ ,Wτ ) ≤ θ−N(τ);

b) if limn→∞|ρ−n (τ)| = 0 then

limn→∞

∥∥I −∆−ρ Σ−(n)

ρ

∥∥∗(Wτ ,Wτ )

= 0.

Proof. (i) a) First we note that the finite matrix ∆+(N)ρ is invertible,

since it is an upper triangle. We get ∆+ρ Σ

+(N)ρ = (anm)n,m≥1 with ann =

1 for all n; an,n+1 = −ρn for all n ≥ N ; and anm = 0 otherwise. For

every X ∈ Wτ , we have (I−∆+ρ Σ

+(N)ρ )X = (ξn(X))n≥1 with ξn(X) = 0

for all n ≤ N − 1 and ξn(X) = ρnxn+1 for all n ≥ N . Then we obtainfor every X ∈ Wτ


∥∥(I −∆+ρ Σ+(N)

ρ )X∥∥

Wτ= sup

n≥N

(1

n

n∑

k=N

|ρkxk+1|τk

)

= supn≥N

(1

n

n∑

k=N

|ρk|τk

τk+1|xk+1|τk+1

)≤ sup

n≥N

[(supk≥N

|ρ+k (τ)|

)1

n

n+1∑

k=N+1

|xk|τk

]

≤ supn≥N

(n + 1

n

)supk≥N

(|ρ+k (τ)|) sup

n≥N

(1

n + 1

n+1∑

k=N+1

|xk|τk

)

≤[(

1 +1

N

)supk≥N

(|ρ+k (τ)|)

]‖X‖Wτ = θ+

N(τ)‖X‖Wτ .

We conclude∥∥I −∆+

ρ Σ+(N)ρ

∥∥∗(Wτ ,Wτ )

≤ θ+N(τ) < 1.

i) b) Part b) is a direct consequence of Part a).ii) Part ii) can be shown similarly. ¤

As a direct consequence of the preceding lemma we get the following

Proposition 3.3. Let ρ, τ ∈ U+.i) a) If ρ+(τ) ∈ `∞ then ∆+

ρ ∈ (Wτ ,Wτ ) and

(6) ‖∆+ρ ‖∗(Wτ ,Wτ ) ≤ 1 + 2‖ρ+(τ)‖`∞ .

b) If

(7) limn→∞

(|ρ+n (τ)|) < 1

then the operator ∆+ρ is a bijection from Wτ to itself.

ii) a) If ρ−(τ) ∈ `∞ then ∆−ρ ∈ (Wτ ,Wτ ) and

‖∆−ρ ‖∗(Wτ ,Wτ ) ≤ 1 + ‖ρ−(τ)‖`∞ ;

b) if

(8) limn→∞

(|ρ−n (τ)|) < 1

then the operator ∆−ρ is a bijection from Wτ to itself and

Wτ (∆−ρ ) = Wτ .

Proof. i) a) We have for each X ∈ Wτ

‖∆+ρ X‖Wτ = sup

n

(1

n

n∑m=1

|xm + ρmxm+1|τm

)


and, since ρmxm+1/τm = ρ+m(τ)xm+1/τm+1, we deduce

‖∆+ρ X‖Wτ ≤ ‖X‖Wτ + sup

m(|ρ+

m(τ)|) supn

(n + 1

n

1

n + 1

n+1∑m=2

( |xm|τm

))

≤ (1 + 2‖ρ+ (τ) ‖`∞

) ‖X‖Wτ ,

and conclude that (6) holds.i) b) By (7), for given l with 0 < l < 1 and for every ε > 0, there is

an integer n0 such that

supn≥n0

(|ρ+n (τ)|) < l + ε.

Then there is an integer n1 with supn≥n1(1 + 1/n) < 1 + ε. So there is

N ≥ max{n0, n1} and a sufficiently small ε > 0 such that

θ+N(τ) ≤ (1 + ε)(l + ε) < 1.

We obtain ∥∥I −∆+ρ Σ+(N)

ρ

∥∥∗(Wτ ,Wτ )

≤ θ+N(τ) < 1,

and ∆+ρ Σ

+(N)ρ has a unique inverse in the Banach algebra (Wτ ,Wτ ).

Since Σ+(N)ρ obviously is bijective from Wτ into itself, the operators

defined by ∆+ρ Σ

+(N)ρ and ∆+

ρ = (∆+ρ Σ

+(N)ρ )(Σ

+(N)ρ )−1 are bijective from

Wτ into itself. So for every given Y ∈ Wτ the equation ∆+ρ X = Y has

a unique solution in Wτ .ii) a) Here we have

‖∆−ρ X‖Wτ = sup

n

(1

n

n∑m=1

|xm + ρmxm−1|τm

)

≤ ‖X‖Wτ + ‖ρ−(τ)‖`∞ supn

(n− 1

n

1

n− 1

n−1∑m=1

|xm|τm

)

≤ (1 + ‖ρ−(τ)‖`∞

) ‖X‖Wτ for all X ∈ Wτ .

This concludes the proof of ii) a).Reasoning as in the proof of i) b), we get ii) b). ¤

4. Spectral properties of an unbounded operatormapping into w∞

In this section we apply the results obtained in the previous sectionto special matrix transformations A from Wξ to w∞, and give somespectral properties of A.


4.1. Definition and first properties of an upper triangle map-ping into w∞. Let a = (an)n≥1 and b = (bn)n≥1 be two sequencesand define the matrix A by [A]nn = an, [A]n,n+1 = bn for all n and[A]nm = 0 otherwise. Then we have

(9) An(X) = anxn + bnxn+1 for all n and all X ∈ s.

We assume that A satisfies the following properties

(10)

{i) a ∈ U+, and there is αA > 0 with an ≥ αAn for all n,ii) there is MA > 0 such that |bn| ≤ MA for all n.

We immediatly get the following properties.

Lemma 4.1. Let a be as in (10). Then we havei) W1/a ⊂ w∞,

ii) W1/a 6= w∞,iii) A ∈ (W1/a, w∞), and for every X ∈ W1/a

(11) ‖AX‖w∞ ≤ ‖X‖W1/a+ 2MA‖X‖w∞ .

Proof. i) Part i) is a consequences of Proposition 3.1.ii) To show Part (ii), let e ∈ w∞ and assume X ′

p = (xnp)n≥1 tends toe in W1/a, that is

‖X ′p − e‖W1/a

= supn

(1

n

n∑m=1

am|xmp − 1|)→ 0 (p →∞).

We have for every n by (10) i)

‖X ′p − e‖W1/a

≥ αA|xnp − 1|and xnp → 1 (p → ∞). There is p0 such that for every p ≥ p0 andevery n, |xnp| ≥ 1/2 and

‖X ′p‖W1/a

≥ 1

n

n∑m=1

am|xmp| ≥ 1

2n

n∑m=1

am.

Again we have for every k by(10) i)

‖a‖w∞ ≥1

2k

2k∑

m=k

am ≥ αA1

2k

2k∑

m=k

m ≥ 3

4αA(k − 1).

Finally we obtain ‖X ′p‖W1/a

≥ ‖a‖w∞/2 = ∞ for p ≥ p0. This contra-dicts the fact that X ′

p ∈ W1/a for all p.iii) Part iii comes from the inequality

1

n

n∑m=1

|bm+1xm+1| = n + 1

n

1

n + 1

n+1∑

k=2

|bkxk| ≤ 2MA‖X‖w∞ for all n.


¤

4.2. Spectral properties of A. We state some elementary lemmaswhich can be found in [4].

Lemma 4.2. Let ε ∈]0, π/2[ and x > 0. Then we have

|x− λ| ≥ x sin ε for all λ ∈ C with |Arg(λ)| ≥ ε.

Lemma 4.3. Let x > 0. Then we have{ |x− λ| ≥ |λ| sin θ for all λ = |λ|eiθ /∈ R−,|x− λ| ≥ |λ| for all λ ∈ R−.

Now we can state the next results on the inverse of A− λI.

Proposition 4.4. Let εA ∈]0, π/2[. Then the infinite matrix A − λIconsidered as operator in W1/a is invertible for every λ ∈ C with|Arg(λ)| ≥ εA, that is

(A− λI)−1 ∈ (W1/a, w∞)

and

(12) ‖(A− λI)−1‖∗(w∞,w∞) ≤M

|λ| for all λ 6= 0 with |Arg(λ)| ≥ εA.

Proof. i) We fix εA ∈]0, π/2[ and consider the sector

ΠεA= {λ ∈ C : |Arg(λ)| < εA}.

We put for every λ /∈ ΠεA

χn =bn

an − λ

and D′λ = D(1/((an−λ)n). Then we have [D′

λ(A − λI)]nn = 1, [D′λ(A −

λI)]n,n+1 = χn and [D′λ(A − λI)]nm = 0 otherwise. We apply Lemma

4.2 with ρ = χ and obtain

|χn| ≤ MA

an sin εA

for all n and all λ /∈ ΠεA

and, since an →∞ (n →∞), there is n0 such that

supn≥n0

|χn| an

an+1

≤ 1

4for all λ /∈ ΠεA

.

Then we get, putting ∆+χ = D′

λ(A− λI) and applying Lemma 3.2

∥∥I −∆+χ Σ+(n0)

χ

∥∥∗(W1/a,W1/a)

≤ θ+n0

(1

a

)≤ 1

4

(1 +

1

n0

)≤ 1

2< 1

for all λ /∈ ΠεA.


Therefore ∆+χ Σ

+(n0)χ is bijective from W1/a to itself and (∆+

χ Σ+(n0)χ )−1 ∈

(W1/a,W1/a). Now we have for every Y ∈ w∞

Y ′ = D ′λY = (yn/(an − λ))n≥1 ∈ W1/a.

Indeed, we get for every n and λ /∈ ΠεA

1

n

n∑m=1

∣∣∣∣ym

am − λ

∣∣∣∣ am ≤ 1

n

n∑m=1

|ym|am sin εA

am ≤ 1

sin εA

‖Y ‖w∞

and ‖Y ′‖W1/a≤ ‖Y ‖w∞/ sin εA. We successively obtain

(∆+

χ Σ+(n0)χ

)−1Y ′ ∈ W1/a,

(A− λI)−1Y = Σ+(n0)χ

(∆+

χ Σ+(n0)χ

)−1Y ′ ∈ W1/a for all Y ∈ w∞

and

(A− λI)−1 = Σ+(n0)χ

(∆+

χ Σ+(n0)χ

)−1D ′

λ ∈ (w∞,W1/a) for all λ /∈ ΠεA.

Now we show that (12) holds. We have, for all λ /∈ ΠεAand for all

Y ∈ w∞, Σ+(n0)χ ∈ (w∞, w∞) and

(13)∥∥(A− λI)−1Y

∥∥w∞

≤∥∥Σ+(n0)

χ

∥∥(w∞,w∞)

∥∥∥(∆+

χ Σ+(n0)χ

)−1∥∥∥

(w∞,w∞)‖D ′

λY ‖w∞ .

It follows that

(14) ‖D ′λY ‖w∞ = sup

n

(1

n

n∑m=1

∣∣∣∣ym

am − λ

∣∣∣∣)≤ sup

n≥1

1

|an − λ|‖Y ‖w∞

and we get by Lemma 4.3

‖D ′λY ‖w∞ ≤ sup

n≥1

1

|an − λ|(15)

≤{

1/|λ| sin θ for λ = |λ|eiθ /∈ R−,

1/|λ| for λ ∈ R−.

Now we have, again by Lemma 3.2

∥∥I −∆+χ Σ+(n0)

χ

∥∥(w∞,w∞)

≤ θ+n0

(e) =

(1 +

1

n0

)supn≥n0

(|χn|) ≤ 1

2


and then we easily get in the Banach algebra (w∞, w∞)

∥∥∥(∆+

χ Σ+(n0)χ

)−1∥∥∥

(w∞,w∞)≤

∞∑m=0

∥∥(I −∆+

χ Σ+(n0)χ

)∥∥m

(w∞,w∞)(16)

≤∞∑

m=0

2−m = 2.

Finally we have from the expression of Σ+(n0)χ in [4, p. 198]

supλ/∈ΠεA

∥∥Σ+(n0)χ

∥∥(w∞,w∞)

< ∞

and we deduce from (13), (14), (15) and (16) that (12) holds. Thisconcludes the proof. ¤

5. Matrix transformations in w∞ and (OGASG)

In this section we apply the previous results to explicitly presentmatrices A and B for which D(A) and D(B) are not embedded in eachother and that are (OGASG).

5.1. Recall of some results in the general case. Here we recallsome results given in Da Prato-Grisvard [1] and Labbas-Terreni [3].The set E is a Banach space and we consider two closed linear operatorsA and B, whose domains are D(A) and D(B) and included in E. Thenwe define SX = AX + BX for every X ∈ D(A) ∩D(B). The spectralproperties of A and B are

(H)

there are CA, CB > 0 and εA, εB ∈]0, π[ such thati) ρ(A) ⊃ ∑

A = {z ∈ C : |Arg(z)| < π − εA} and

‖(A− zI)−1‖£(E) ≤CA

|z| for all z ∈ ∑A−{0};

ii) ρ(B) ⊃ ∑B = {z ∈ C : |Arg(z)| < π − εB} and

‖(B − zI)−1‖£(E) ≤CB

|z| for all z ∈ ∑B −{0};

iii) εA + εB < π

If (H) is satisfied then A and B are (OGASG) not strongly continuousat t = 0 and we have σ(A)∩ σ(−B) = ∅ and ρ(A)∪ ρ(−B) = C. In [4]an application was given for the solvability of the equation (A + B +λI)X = Y where A and B were considered as operators in `∞ in thenoncommutative case.


Note that it is well known (cf. [1]) in the commutative case, that iswhen

(A− ξI)−1(B − ηI)−1 − (B − ηI)−1(A− ξI)−1 = 0

for all ξ ∈ ρ(A), η ∈ ρ(B),

that if D(A) and D(B) are dense in E, then the bounded operatordefined by

Lλ = − 1

2iπ

∫

Γ

(B + zI)−1(A− λI − zI)−1 dz for all λ > 0,

where Γ is an infinite sectorial curve in ρ(A − λI) ∩ ρ(−B), coincideswith (A + B − λI)−1.

5.2. Matrix transformations as (OGASG). In our case A is de-fined in Subsection 4.1 by (9) and (10). Then the matrix B is definedfor β = (βn)n≥1, γ = (γn)n≥1 ∈ s by

Bn(X) = γnxn−1 + βnxn for all n and all X ∈ s

with the convention x0 = 0, where we assume

(17)

i) β ∈ U+ and limn→∞ β2k = L 6= 0,ii) limk→∞ β2k+1/a2k+1 = ∞iii) α) there is MB > 0 such that |γ2k| ≤ MB for all n,

β) γ2k+1 = o(1) (n →∞).

We easily see that B ∈ (W1/β, w∞) and for each X ∈ W1/β

‖BX‖w∞ ≤ ‖X‖W1/β+ MB‖X‖w∞ .

These results lead to the next remarks.

Remark 5.1. We note for the convenience of the reader that, for in-stance, we can define A and B as follows; an = n; b = e; βn = 1 ifn = 2k, βn = k2 if n = 2k+1; γn = 1 if n = 2k, γn = 1/k if n = 2k+1for all n and k.

Remark 5.2. We will see in Lemma 5.3 and Theorem 5.4 that the con-dition in (10) ii) implies that A is a closed operator. The conditions in(10) i) and (17) i), ii) imply that D(A) and D(B) are not embedded ineach other. We will see in Theorem 5.4 that the conditions in (10) i)and (17) ii), iii) imply that B + µI considered as an operator in W1/β

is invertible and

(B + µI)−1 ∈ (w∞,W1/β) for all µ with |Arg(µ)| ≤ π − εB.

Now we state the next result.


Lemma 5.3. Let A, B be as in (10) and (17). Then we havei) W1/a ⊂ w∞, W1/β ⊂ w∞,ii) w∞(A) = W1/a and w∞(B) = W1/β,iii) W1/a and W1/β are not embedded in each other,

iv) W1/a 6= w∞, W1/β 6= w∞.

Proof. i) The inclusion W1/a ⊂ w∞ follows from Proposition 3.1 whereτ = 1/a, ν = e and 1/a ∈ c0 ⊂ `∞. Furthermore we have 1/β2k+1 =(1/a2k+1)(a2k+1/β2k+1) = o(1) (k →∞) and W1/β ⊂ w∞.

ii) First we have A ∈ (W1/a, w∞) by Lemma 4.1. It remains toshow w∞(A) = W1/a. First we have W1/a ⊂ w∞(A). Indeed it followsfrom [13, Theorem 1, pp. 260] that I ∈ (W1/a, w∞(A)) if and only ifA ∈ (W1/a, w∞). Now let X ∈ w∞(A). Then we have Y = AX ∈ w∞.In the proof of Proposition 4.4, we can take λ = 0. Indeed there is n0

such that

χn =|bn|an

≤ MA

nαA

for all n ≥ n0.

Then, for Y ∈ w∞, we successively get D1/aY = (yn/an)n≥1 ∈ W1/a,

(∆+χ Σ

+(n0)χ )−1D1/aY ∈ W1/a, and since A−1 = Σ

+(n0)χ (∆+

χ Σ+(n0)χ )−1D1/a,

we concludeX = A−1Y ∈ W1/a.

This shows w∞(A) ⊂ W1/a, and since W1/a ⊂ w∞(A), we concludew∞(A) = W1/a. The proof is similar for B.

iii) Part iii) is a direct consequence of Proposition 3.1, since a/β andβ/a /∈ `∞.

iv) The property W1/a 6= w∞ has been shown in Lemma 4.1. To

show W1/β 6= w∞, we use the notations of Lemma 4.1. Here we have

(18)∥∥X ′

p − e∥∥

W1/β= sup

n

(1

n

n∑m=1

βm|xmp − 1|)→ 0 (p →∞)

and

∥∥X ′p − e

∥∥W1/β

≥ supn

1

n

n−12∑

k=1

β2k+1|x2k+1,p − 1|

≥ supn

1

n

n−12∑

k=1

β2k+1

a2k+1

a2k+1|x2k+1,p − 1|

≥ αA supn

1

n

n−12∑

k=1

β2k+1

a2k+1

(2k + 1)|x2k+1,p − 1| .


Since β2k+1/a2k+1 →∞ (k →∞), there is C1 > 0 with β2k+1/a2k+1 ≥C1 for all k and

∥∥X ′p − e

∥∥W1/β

≥ αAC1

nn|xnp − 1| = αAC1|xnp − 1| for all n.

From (18), we have xnp → 1 (p → ∞) for all n. There is p0 such thatfor every n and each p ≥ p0, |xnp| ≥ 1/2 and

(19)∥∥X ′

p

∥∥W1/β

≥ 1

2n

n∑m=1

βm =1

2‖β‖w∞ .

We have for every integer i

‖β‖w∞ ≥1

4i + 3

2i+1∑

k=1

β2k+1 =1

4i + 3

2i+1∑

k=1

β2k+1

a2k+1

a2k+1

≥ αAC1

4i + 3

2i+1∑

k=1

(2k + 1) ≥ αAC1(2i + 1)(2i + 3)

4i + 3.

Since (2i + 1)(2i + 3)/(4i + 3) →∞ (i →∞), we deduce ‖β‖w∞ = ∞,and by (19) we get ‖X ′

p‖W1/β≥ ‖β‖w∞/2 = ∞ for p ≥ p0. This

contradicts the fact that X ′p ∈ W1/β for all p. Therefore we conclude

W1/β 6= w∞. ¤

We immediatly obtain the next result.

Theorem 5.4. The two linear operators A and B are closed in thespace w∞ and satisfy

i) D(A) = W1/a,ii) D(B) = W1/β,

iii) D(A) 6= w∞, D(B) 6= w∞.iv) There are εA, εB > 0 (with εA + εB < π) such that (12) holds

and

(20)∥∥(B + µI)−1

∥∥∗(w∞,w∞)

≤ M

|µ| for all µ 6= 0 and |Arg(µ)| ≤ π−εB.

Proof. We show that A is a closed operator. For this, we consider asequence X ′

p = (xnp)n≥1 tending to X = (xn)n≥1 in w∞ as p tends toinfinity, where X ′

p ∈ W1/a for all p. Then we have AX ′p → Y (p →∞)

in w∞ with Y = (yn)n≥1. It follows that for every n

An(X ′p) → An(X) = yn (p →∞)


with An(X ′p) = anxnp + bnxn+1,p, yn = anxn + bnxn+1, and since

1

n

n∑m=1

|amxm| = 1

n

n∑m=1

|ym − bmxm+1|

≤ ‖Y ‖w∞ + 2MA‖X‖w∞ for all n,

we conclude X ∈ W1/a. The proof for B is similar.i), ii) and iii) These parts are direct consequences of Lemma 5.3.iv) The first part in iv) has been shown in Proposition 4.4. Let

εB ∈]0, π/2[. We show that for every µ with |Arg(µ)| ≤ π − εB, theinfinite matrix B + µI, considered as an operator in W1/β, is invertibleand

(B + µI)−1 ∈ (W1/β, w∞).

We put ΣB = {µ ∈ C : |Arg(µ)| ≤ π − εB}. To be able to deal withthe inverse of B + µI, we need to study the sequences with |γ2k+1|/β2k

and |γ2k/(β2k + µ)|β2k/β2k−1. We have by (17) i), iii)

(21) γ2k+1/β2k → 0 (k →∞).

On the other hand we get for every µ ∈ ΣB

(22)

∣∣∣∣γ2k

β2k + µ

∣∣∣∣β2k

β2k−1

≤ MB

β2k sin εB

β2k

β2k−1

=MB

sin εB

1

β2k−1

and

(23)1

β2k−1

=a2k−1

β2k−1

1

a2k−1

= o(1) (k →∞).

We deduce from (21), (22) and (23) that there is n1 such that

|γ2k+1| 1

β2k

≤ 1

4sin εB for 2k + 1 ≥ n1,

and ∣∣∣∣γ2k

β2k + µ

∣∣∣∣β2k

β2k−1

≤ 1

4for 2k ≥ n1 for all µ ∈ ΣB.

We define the matrices D ′µ = D(1/(βn+µ)n), and then D ′

µ(B + µI) = ∆−κ

with κn = γn/(βn + µ). We have

σ1 = |κ2k| β2k

β2k−1

≤ MB

β2k sin εB

β2k

β2k−1

≤ 1

4for all k ≥ n1 − 1

2

and

σ2 = |κ2k+1|β2k+1

β2k

≤ 1

4sin εB

1

β2k+1

sin εBβ2k+1 =1

4

for all k ≥ n1

2.


It follows that

supn≥n1

(|κn| βn

βn−1

)= max{σ1, σ2 max} ≤ 1

4for all µ ∈ ΣB.

Lemma 3.2 yields

∥∥I −∆−κ Σ(n1−1)

κ

∥∥∗(W1/β ,W1/β)

≤ θ−n1

(1

β

)≤ 1

4

(1 +

1

n1 − 1

)≤ 1

2< 1

for all µ ∈ ΣB

So ∆−κ = ∆−

κ Σ(n1−1)κ (Σ

(n1−1)κ )−1 is bijective from W1/β to itself. Reason-

ing as in Proposition 4.4 with (B + µI)−1 = (∆−κ )−1D ′

µ, we concludethat B+µI, considered as an operator from W1/β into w∞ is invertible,and (B + µI)−1 ∈ (w∞,W1/β) for all µ ∈ ΣB. Condition (20) can beobtained by reasoning as in Proposition 4.4. This concludes the proofof Part iv). ¤

References

[1] G. Da Prato, P. Grisvard, Sommes d’operateurs lineaires et equations dif-ferentielles operationnelles. J. Math. Pures Appl. IX Ser. 54 (1975), 305-387.

[2] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.[3] R. Labbas, B. Terrini, Sommes d’operateurs lineaires de type parabolique, 1ere

partie. Boll. Un. Mat. Italiana, 7, 1-B (1987), 545-569.[4] R. Labbas, B. de Malafosse, An application of the sum of linear operators in

infinite matrix theory, Fac. des Sci. de l’Univ. d’Ank., Ser. Al Math. and stat.46 (1997), 191-210.

[5] I. J. Maddox, On Kuttner’s theorem, J. London Math. Soc. 43, 1968, (285-290).[6] B. de Malafosse, Application of the sum of operators in the commutative case

to the infinite matrix theory, Soochow J. of Math. 27, N◦.4, (2001), 405-421.[7] B. de Malafosse, Some properties of the sum of linear operators, Novi Sad

Journal of Mathematic 33, 1 (2003), 75-91.[8] B. de Malafosse, On some BK space, Int. J. of Math. and Math. Sc. 28 (2003),

1783-1801.[9] B. de Malafosse, The Banach algebra B (X) , where X is a BK space and

applications, Vesnik math. journal, 57 (2005), 41-60.[10] B. de Malafosse, E. Malkowsky, The Banach algebra (w∞ (λ) , w∞ (λ)), ac-

cepted for publication in FJMS, Far East Journal of Mathematical Sciences(2005).

[11] B. de Malafosse, V. Rakocevic, Applications of measure of noncompactness inoperators on the spaces sα, s0

α, s(c)α and `p

α, J. Math. Anal. Appl. 323 (2006),131-145.

[12] E. Malkowsky, The continuous duals of the spaces c0(Λ) and c(Λ) for expo-nentially bounded sequences Λ , Acta Sci. Math. (Szeged) 61 (1995), 241-250.

[13] E. Malkowsky, Linear operators in certain BK spaces, Bolyai Soc. Math. Stud.5 (1996), 259-273.


[14] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spacesand measure of noncompactness, Zbornik radova, Matematicki institut SANU9 (17) (2000), 143-243.

[15] A. Wilansky, Summability through Functional Analysis, North-Holland Math-ematics Studies 85, 1984.

(de Malafosse) LMAH Universite du Havre, BP 4006 I.U.T Le Havre,76610 Le Havre. France.

(Malkowsky) Mathematisches Institut, Universitat Gießen, Arndt-str. 2, D–35392 Gießen, Germany, c/o School of Informatics and Com-puting, German–Jordanian University, P.O. Box 35247, Amman, Jordan

E-mail address, de Malafosse: [email protected] address, Malkowsky: [email protected] address: [email protected]


BEST SIMULTANEOUS APPROXIMATION IN METRIC SPACES

SH. AL-SHARIF

Abstract. For a Banach space X and an increasing subadditive continuous functionϕ on [0,∞) with ϕ(0) = 0, let us denote by L

ϕ

(I,X), the space of all X-valued ϕ-integrable functions f : I → X on a certain positive complete σ-finite measure space

(I,∑

, µ, ) with∫I

ϕ ‖f(t)‖ dµ(t) < ∞ and lϕ

(X) ={

(xk) :∞∑

k=1

ϕ ‖xk‖ < ∞, xk ∈ X

}.

The aim of this paper is to prove that for a closed separable subspace G of X, Lϕ

(I,G)is simultaneously proximinal in L

ϕ

(I,X) if and only if G is simultaneously proximinalin X. Other result on simultaneous approximation of l

ϕ

(G) in lϕ

(X) is presented.

1. Introduction

A function ϕ : [0,∞) → [0,∞) is called a modulus function if it satisfies the followingconditions:

(1) ϕ is continuous and increasing function.(2) ϕ(x) = 0 if and only if x = 0.(3) ϕ(x+ y) ≤ ϕ(x) + ϕ(y).The functions ϕ(x) = xp, 0 < p < 1, and ϕ(x) = ln(1 + x) are modulus functions.

In fact if ϕ is a modulus function, then ψ(x) = ϕ(x)/ (1 + ϕ(x)) is a modulus function.Further the composition of two modulus function is a modulus function.

For a modulus function ϕ and a Banach space X, let us denote by Lϕ(I,X), the space

of all X-valued ϕ-integrable functions f : I → X on a certain positive complete σ-finitemeasure space (I,

∑, µ, ) with

∫I

ϕ ‖f(t)‖ dµ(t) <∞ and

lϕ

(X) =

{(xk) :

∞∑k=1

ϕ ‖xk‖ <∞, xk ∈ X

}.

For a = (ak) ∈ lϕ(X) and f ∈ Lϕ

(I,X) set

‖a‖ϕ

=∞∑

k=1

ϕ ‖ak‖ and ‖f‖ϕ

=

∫I

ϕ ‖f(t)‖ dµ(t).

If X = C, the set of complex numbers, the spaces lϕ(X) and L

ϕ(I,X) is simply

denoted by lϕ

and Lϕ(I) respectively. It is known, [4] , that l

ϕ ⊆ l1, Lϕ(I) ⊇ L1(I)

1991 Mathematics Subject Classification. Primary: 41A65; Secondary: 41A50.Key words and phrases. Simultaneous, Approximation.Copyright c© Deanship of Research and Graduate Studies, Yarmouk University, Irbid, Jordan.

Received Sept. 1, 2007, Accepted March 4, 2008.69

70 SH. AL-SHARIF

and(lϕ(X), ‖.‖

ϕ

)and

(L

ϕ(I,X), ‖.‖

ϕ

)are complete metric linear spaces. For more on

lϕ

and Lϕ(I) we refer to the reader to [3] and [5] .

Note that the Banach space X is a metric space with the metric d(x, y) = ϕ ‖x− y‖ .

Definition 1.1. Let ϕ be a modulus function and G be a closed subspace of a Banachspace X. We say that

(a) G is simultaneously proximinal in X if for each m-tuple of elements(x1, x2, ..., xm) ∈ Xm there exists g ∈ G such that:

m∑i=1

ϕ ‖xi − g‖ = distϕ(x1, x2, ..., xm, G) = infh∈G

m∑i=1

ϕ ‖xi − h‖ .

In other words for every h ∈ G

‖(x1, x2, ..., xm, 0, ...)− (g, g, ..., g, 0, ...)‖ϕ≤

∥∥∥∥ (x1, x2, ..., xm, 0, ...)− (h, h, ..., h, 0, ...)

∥∥∥∥ϕ

.

(b) Lϕ(I,G) is simultaneously proximinal in L

ϕ(I,X) if for each m-tuple of elements

f1, f2, ..., fm ∈(L

ϕ(I,X)

)mthere exists g ∈ Lϕ

(I,G) such that

m∑i=1

‖fi − g‖ϕ

= distϕ(f1, f2, ..., fm, Lϕ

(I,G)) = infh∈L

ϕ(I,G)

m∑i=1

‖fi − h‖ϕ.

The problem of best simultaneous approximation has been studied by many authorse.g., [2] , [9] , [15] and [16] . Most of these works have dealt with the characterizationof best simultaneous approximation in spaces of continuous functions with values in aBanach space X. Some existence and uniqueness results were obtained. Results on bestsimultaneous approximation in general Banach spaces may be found in [11] and [13] .Related results on Lp(I,X), 1 ≤ p <∞, are given in [14] . In [14] , it is shown that if G isa reflexive subspace of a Banach space X, then Lp(I,G) is simultaneously proximinal inLp(I,X). If p = 1, Abu Sarhan and Khalil [1] , proved that if G is a reflexive subspace ofthe Banach space X or G is a 1-summand subspace of X, then L1(I,G) is simultaneouslyproximinal in L1(I,X).

The aim of this paper is to prove that for a closed separable subspace G of X, Lϕ(I,G)

is simultaneously proximinal in Lϕ(I,X) if and only if G is simultaneously proximinal

in X. Some results are inspired by the results in [14] . Other result on simultaneousapproximation of l

ϕ(G) in l

ϕ(X) is presented.

Throughout this paper, (I,∑, µ, ) is a σ-finite measure space, X is a Banach space,

G is a closed subspace of X and the norm of v ∈ X is denoted by ‖v‖ .

2. Distance Formulae

Progress in the discussion of simultaneous proximinality when X does not possesspleasant properties is greatly facilitated by the fact that the distance from an m-tupleof elements f1, f2, ..., fm ∈ L

ϕ(I,X) to a subspace L

ϕ(I,G) is computed by the following

theorem:

BEST SIMULTANEOUS APPROXIMATION IN METRIC SPACES 71

Theorem 2.1. Let ϕ be a modulus function and f1, f2, ..., fm ∈ Lϕ(I,X). Then

distϕ

(f1, f2, ..., fm, L

ϕ

(I,G))

=

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s).

Proof. Let f1, f2, ..., fm ∈ Lϕ(I,X). Then for each i = 1, 2, ...,m, fi is the limit al-

most everywhere of a sequence of simple functions {fi,n} in Lϕ(I,X).Since the distance

function distϕ(x,G) is continuous in x ∈ X, limn→∞

ϕ (‖fi,n(s)− fi(s)‖) = 0, i = 1, 2, ...,m,

implies that

limn→∞

∣∣distϕ (f1,n(s), ..., fm,n(s), G))− distϕ (f1(s), ..., fm(s), G))∣∣ = 0.

Furthermore for each n, the function: s→distϕ (f1,n(s), f2,n(s), ..., fm,n(s), G) is a simplefunction and so we may assume that distϕ (f1(s), f2(s), ..., fm(s), G)) is measurable. Nowfor any g ∈ Lϕ

(µ,G)

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s) ≤∫I

m∑i=1

ϕ (‖fi(s)− g(s)‖) dµ(s)

=m∑

i=1

∫I

ϕ (‖fi(s)− g(s)‖) dµ(s).

Therefore

(1)

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s) ≤ distϕ

(f1, f2, ..., fm, L

ϕ

(I,G)).

For the reverse inequality fix ε > 0. Since simple functions are dense in Lϕ(I,X), there

exist simple functions, f′j in L

ϕ(I,X) such that

∥∥fj − f′j

∥∥ϕ< ε

m, j = 1, 2, ...,m. Assume

that f′j(t) =

n∑i=1

κAi

(t)yji , j = 1, 2, ...,m, where κ

Aiare the characteristic functions of

the measurable sets Ai in I and yji ∈ X. We can assume that

n∑i=1

κAi

= 1 and µ(Ai) > 0.

Given ε > 0 for each i = 1, 2, ..., n, select gi ∈ G such that:

m∑j=1

ϕ∥∥yj

i − gi

∥∥ < distϕ(y1i , y

2i , ..., y

mi , G) +

ε

nµ(Ai).

72 SH. AL-SHARIF

Let g(t) =n∑

i=1

κAi

(t)gi. Clearly g ∈ Lϕ(I,G) and

distϕ(f1, ..., fm, L

ϕ

(I,G))≤

m∑j=1

∥∥∥fj − f′

j

∥∥∥ϕ

+ distϕ

(f

′

1, f′

2, ..., f′

m, Lϕ

(I,G))

≤ ε+m∑

j=1

∥∥∥f ′

j − g∥∥∥

ϕ

= ε+m∑

j=1

∫I

ϕ∥∥∥f ′

j(s)− g(s)∥∥∥ dµ(s)

= ε+m∑

j=1

n∑i=1

∫Ai

ϕ∥∥∥f ′

j(s)− g(s)∥∥∥ dµ(s)

= ε+m∑

j=1

n∑i=1

(ϕ

∥∥∥yj

i − gi

∥∥∥)µ(Ai)

= ε+n∑

i=1

m∑j=1

(ϕ

∥∥∥yj

i − gi

∥∥∥)µ(Ai)

≤ ε+n∑

i=1

µ(Ai)distϕ(y1i , y

2i ..., y

mi , G) +

ε

n

≤ 2ε+n∑

i=1

∫Ai

distϕ(y1i , y

2i , ..., y

mi , G)dµ(s)

= 2ε+

∫I

distϕ

(f

′

1(s), f′

2(s), ..., f′

m(s), G)dµ(s).

Since

distϕ

(f

′

1(s), f′

2(s), ..., f′

m(s), G)≤ distϕ (f1(s), f2(s), ..., fm(s), G)

+m∑

j=1

ϕ∥∥∥f ′

j(s)− f(s)∥∥∥


then,

distϕ(f1, f2, ..., fm, L

ϕ

(I,G))≤ 2ε+

m∑j=1

∫I

ϕ∥∥∥f ′

j(s)− fj(s)∥∥∥ dµ(s)

+

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s)

= 2ε+m∑

j=1

∥∥∥fj − f′

j

∥∥∥ϕ

+

∫I

distϕ(f1(s), f2(s), ..., fm(s), G)dµ(s)

≤ 3ε+

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s),

which (since ε is arbitrary) implies that

(2) distϕ(f1, f2, ..., fm, L

ϕ

(I,G))≤

∫I

distϕ (f1(s), f2(s), ..., fm(s), G) dµ(s).

Hence by 1 and 2 the proof is complete. �

An application of Theorem 2.1 is

Corollary 2.2. An element g ∈ Lϕ(I,G) is a best simultaneous approximation of

f1, f2, ..., fm ∈ Lϕ(I,X) if and only if g(t) is a best simultaneous approximation of

f1(t), f2(t), ..., fm(t) ∈ X for almost all t ∈ I.

3. Best Simultaneous Approximation in Lϕ(I,X)

The main result in this section is, for a modulus function ϕ and a closed separablesubspace G of a Banach space X, L

ϕ(I,G) is simultaneously proximinal in L

ϕ(I,X) if

and only if G is simultaneously proximinal in X. We begin with the following:

Theorem 3.1. If G is simultaneously proximinal in X, then for every m-tuple ofsimple function f1, f2, ..., fm ∈ L

ϕ(I,X), P (f1, f2, ..., fm, L

ϕ(I,X)) is not empty, where

P (f1, f2, ..., fm, Lϕ(I,X)) is the set of all elements g ∈ L

ϕ(I,G) such that g is a best

simultaneous approximation of m-tuple of the elements f1, f2, ..., fm.

Proof. Let f1, f2, ..., fm be an m-tuple of simple functions in Lϕ(I,X). With no loss of

generality we can assume that fj(t) =n∑

i=1

κAi

(t)yji , where Ai are disjoint measurable sets

such thatn⋃

i=1

Ai = I. Pick gi ∈ G such that gi is a best simultaneous approximation of

74 SH. AL-SHARIF

the m-tuple of elements y1i , y

2i , ..., y

mi ∈ X, i = 1, 2, ..., n. Set g(t) =

n∑i=1

κAi

(t)gi. Then for

any h ∈ Lϕ(I,X) we have:

m∑j=1

‖fj − h‖ϕ

=m∑

j=1

∫I

ϕ ‖fj(s)− h(s)‖ dµ(s)

=

∫I

m∑j=1

ϕ ‖fj(s)− h(s)‖ dµ(s)

=n∑

i=1

∫Ai

m∑j=1

ϕ∥∥yj

i − h(s)∥∥ dµ(s)

≥n∑

i=1

∫Ai

m∑j=1

ϕ∥∥yj

i − gi

∥∥ dµ(s)

=

∫I

m∑j=1

ϕ ‖fj(s)− g(s)‖ dµ(s).

Hencem∑

j=1

‖fj − g‖ϕ

= infh∈L

ϕ(I,G)

m∑j=1

‖fj − h‖ϕ

�

Theorem 3.2. If ϕ is a modulus function, then G is simultaneously proximinal in Xif L


ϕ(I,X).

Proof. Let x1, x2, ..., xm ∈ X. Set fj = 1 ⊗ xj, j = 1, 2, ...,m, where 1 is the constantfunction 1. Clearly for each j = 1, 2, ...,m, fj ∈ L

ϕ(I,X). By assumption there exists

g ∈ Lϕ(I,G) such that for any h ∈ Lϕ

(I,G)

m∑j=1

‖fj − g‖ϕ≤

m∑j=1

‖fj − h‖ϕ.

By Theorem 2.1m∑

j=1

ϕ ‖fj(t)− g(t)‖ ≤m∑

j=1

ϕ ‖fj(t)− h(t)‖

a.e. in I. Orm∑

j=1

ϕ ‖xj − g(t)‖ ≤m∑

j=1

ϕ ‖xj − h(t)‖ .

Let h run over all functions 1⊗ z, for z ∈ G, we getm∑

j=1

ϕ ‖xj − g(t)‖ ≤m∑

j=1

ϕ ‖xj − z‖ .

�


Now we pose the following problem: If G is separable is it true thatL


ϕ(I,X)? to solve this problem we

begin by the following:

Lemma 3.3. [Lemma 2.9 of [9]] Assume µ(I) < +∞. Suppose (M,d) is a metric spaceand A is a subset of I such that µ∗(A) = µ(I), where µ∗ denotes the outer measureassociated to µ. If g is a mapping from I to M with separable range, then for any ε > 0there exists a countable partition {En} of I in measurable sets and An ⊂ A ∩ En suchthat µ∗(An) = µ(En) and diam(g(An)) < ε for all n.

Theorem 3.4. Let G be a closed separable subspace of X. Let us suppose that G issimultaneously proximinal in X and f1, f2, ..., fm : I → X be measurable functions.Then there is a measurable function g : I → X such that g(t) is a best simultaneousapproximation of (f1(t), f2(t), ..., fm(t)) in G for almost all t.

Proof. Let f1, f2, ..., fm : I → X be measurable functions. So we may assume thatf1(I), f2(I), ..., fm(I) are separable sets in X. Using the fact that µ is σ-finite we canfind countable partitions {I1n}∞n=1 , {I2n}∞n=1 , ..., {Imn}∞n=1of I in measurable sets suchthat diamϕ(fi(Iin) < 1

2and µ(Iin) <∞, i = 1, 2, ...,m, for all n, where

diamϕA = sup {ϕ ‖x− y‖ : x, y ∈ A} .

Consider the partition {In1,n2,...,nm}∞, mni=1,i=1 , where In1,n2,...,nm =

m⋂i=1

Iin, for 1 ≤ ni < ∞.

Then diamϕ(fi(In1,n2,...,nm)) < 12, i = 1, 2, ...,m. For simplicity we write {In1,n2,...,nm}

∞,mni,i=1

as {In}∞n=1 . For each t ∈ I, let g0(t) be a best simultaneous approximation of(f1(t), f2(t), ..., fm(t)) in G. Define g0 from I into G such that g0(t) is a best simul-taneous approximation of (f1(t), f2(t), ..., fm(t)) . Applying Lemma 3.3 to the mappingg0 in each In taking ε = 1

2and I = A = In. We get a countable partition in each In

and therefore a countable partition in the whole of I. Thus we get a countable partition{En}∞n=1 of I in measurable sets and a sequence of subsets {An}∞n=1 of I such that

An ⊆ En, µ∗ (An) = µ (En) < +∞,

diamϕ(g0(An)) <1

2, diam

ϕ(fi(En)) <

1

2, i = 1, 2, ...,m.

Let us apply again the same argument in each En with ε = 122 , I = En and A = An .

For each n we get a countable partition {Enk: 1 ≤ k <∞}of En in measurable sets and

a sequence {Ank: 1 ≤ k <∞} of subsets of I such that

Ank⊆ Enk

∩ An, µ∗ (Ank

) = µ (Enk) ,

diamϕ(g0(Ank)) <

1

22and diamϕ(fi(Enk

)) <1

22, i = 1, 2, ...,m,

for all n and k. Let us proceed by induction. Now for each natural number k, let 4k

be the set of k-tuples of natural numbers and let 4 =∞⋃

k=1

4k. On this 4 consider the

partial order defined by (m1,m2, ...,mi) ≤ (n1, n2, ..., nj) if and only if i ≤ j and mk = nk

76 SH. AL-SHARIF

for k = 1, 2, ..., i. Then by induction for each natural number k, we can take a partition{Eα : α ∈ 4k} of subsets of I and a collection {Aα}α∈4k

such that:

(1) Aα ⊆ Eα and µ∗ (Aα) = µ (Eα) for each α.(2) Aα ⊆ A

βand Eα ⊆ E

βif β ≤ α.

(3) diamϕ(fi(Eα)) < 12k for i = 1, 2, ...,m and diamϕ(g0(Aα)) < 1

2k if α ∈ 4k.

We may assume that Aα 6= ∅ for all α ( forget the α′s for which Aα = ∅). For each

α ∈ 4 take tα ∈ Aα and define gk from I into G by gk(.) =∑

α∈4k

κEα

(.)g0(tα). Then for

each t ∈ I and n ≤ k we have:

ϕ ‖gn(t)− gk(t)‖ = ϕ

∥∥∥∥∥ ∑α∈4n

κEα

(t)g0(tα)−∑

β∈4k

κE

β(t)g0(tβ)

∥∥∥∥∥ .But since n ≤ k by 1 and 2 we have:

ϕ ‖gn(t)− gk(t)‖ ≤ ϕ

∥∥∥∥∥ ∑β∈4k

κE

β(t)

(g0(tα)− g0(tβ)

)∥∥∥∥∥≤

∑β∈4k

φ∥∥(g0(tα)− g0(tβ)

)∥∥µ(Eβ)

≤ 1

2n.

Therefore (gk(t)) is a Cauchy sequence in X for every t ∈ I. Consequently (gk(t)) is aconvergent sequence for every t ∈ I. Let g : I → G be the point wise limit of (gk). Sincegk is measurable for each k, g is measurable. Let t ∈ I and let n be a natural number.Suppose t ∈ Eα . We have:

m∑i=1

ϕ ‖fi(t)− gn(t)‖ =m∑

i=1

ϕ ‖fi(t)− g0(tα)‖

≤m∑

i=1

ϕ ‖fi(t)− fi(tα)‖+ ϕ ‖fi(tα)− g0(tα)‖

≤m∑

i=1

1

2n+ ϕ ‖fi(tα)− g0(tα)‖

≤ m

2n+ distϕ((f1(tα), f2(tα), ..., fm(tα)), G)

≤ m

2n+

m∑i=1

ϕ ‖fi(t)− fi(tα)‖

+distϕ((f1(t), f2(t), ..., fm(t)), G)

≤ m

2n−1+ distϕ((f1(t), f2(t), ..., fm(t)), G).


Letting n→∞ we get:

m∑i=1

ϕ ‖fi(t)− g(t)‖ = limn→∞

m∑i=1

ϕ ‖fi(t)− gn(t)‖

= distϕ((f1(t), f2(t), ..., fm(t)), G).

and so g(t) is a best simultaneous approximation of f1(t), f2(t), ..., fm(t) in G. �

Theorem 3.5. Let ϕ be a modulus function and G be a closed separable subspace of X.Then L


ϕ(I,X) if and only if G is simultane-

ously proximinal in X.

Proof. Necessity is in Theorem 3.2 Let us show sufficiency. Suppose that G is simul-taneously proximinal in X, and let f1, f2, ..., fm be functions in L

ϕ(I,X). Theorem 3.4

guarantees that there exists a measurable function g defined on I with values in X suchthat g(t) is a best simultaneous approximation of f1(t), f2(t), ...fm(t) in G for almost all t.It follows from Corollary 2.2 that g is a best simultaneous approximation of f1, f2, ..., fm

in Lϕ(I,G) �

Theorem 3.6. Let ϕ be a modulus function. Then if g ∈ Lϕ(I,G) is a best simultaneous

approximation from Lϕ(I,G) of an m-tuple of elements f1, f2, ..., fm ∈ L

ϕ(I,X) then for

every measurable subset A of I and every h ∈ Lϕ(I,G),

∫A

ϕ (‖fj0(s)− g(s)‖) dµ(s) ≤∫A

ϕ (‖fj0(s)− h(s)‖) dµ(s),

for some j0 ∈ {1, 2, ...,m} .

Proof. If µ(A) = 0 then there is nothing to prove. Suppose that for some A satisfyingµ(A) > 0 and for some h0 ∈ L

ϕ(I,G), the inequality does not hold for J = 1, 2, ...,m.

Now, define g0 ∈ Lϕ(I,G) by

g0(s) :=

{g(s) if s ∈ I − Ah0(s) if s ∈ A

78 SH. AL-SHARIF

Then we have for j = 1, 2, ...,m∫I

ϕ (‖fj(s)− g0(s)‖) dµ =

∫A

ϕ (‖fj(s)− h0(s)‖) dµ(s)

+

∫I−A

ϕ (‖fj(s)− g(s)‖) dµ(s)

<

∫A

ϕ (‖fj(s)− g(s)‖) dµ(s)

+

∫I−A

ϕ (‖fj(s)− g(s)‖) dµ(s)

=

∫I

ϕ (‖fj(s)− g(s)‖) dµ(s).

This implies thatm∑

j=1

‖fj − g0‖ϕ<

m∑j=1

‖fj − g‖ϕ

which contradict the fact that g is a best simultaneous approximation from Lϕ(I,G) of

the m-tuple of elements f1, f2, ..., fm. �

As a corollary we get:

Corollary 3.7. If g is a best simultaneous approximation from Lϕ(I,G) of an m-tuple

of elements f1, f2, ..., fm ∈ Lϕ(I,X) then, for every measurable subset A if I,∫

A

ϕ (‖g(s)‖) dµ(s) ≤ 2 max1≤j≤m

∫A

ϕ (‖fj(s)‖) dµ(s)

.

Proof. Since, for j = 1, 2, ...,m∫A

ϕ (‖g(s)‖) dµ(s) ≤∫A

ϕ (‖fj(s)− g(s)‖) dµ(s) +

∫A

ϕ (‖fj(s)‖) dµ(s),

we obtain, by using Theorem 3.6 with h =: 0, that for j0 ∈ {1, 2, ...,m}∫A

ϕ (‖g(s)‖) dµ(s) ≤ 2

∫A

ϕ (‖fj0(s)‖) dµ(s)

≤ 2 max1≤j≤m

∫A

ϕ (‖fj(s)‖) dµ(s)

,

which completes the proof. �


We end this paper with the following result on best simultaneous approximation oflϕ(X) in l

ϕ(G).

Theorem 3.8. Let ϕ be a modulus function. Then lϕ(G) is simultaneously proximinal

in lϕ(X) if G is simultaneously proximinal in X.

Proof. Let f1, f2, ..., fm ∈ lϕ(X). Since G is simultaneously proximinal in X, for each

n, there exists g(n) ∈ G such that for every y ∈ Gm∑

j=1

ϕ ‖fj (n)− g(n)‖ ≤m∑

j=1

ϕ ‖fj (n)− y‖ .

Since y = 0 ∈ G, we getm∑

j=1

ϕ ‖fj (n)− g(n)‖ ≤m∑

j=1

ϕ ‖fj (n)‖ .

But ϕ is increasing and subadditive so

mϕ ‖g(n)‖ =m∑

j=1

ϕ ‖g(n)− fj (n) + fj (n)‖

≤m∑

j=1

ϕ ‖g(n)− fj (n)‖+ ϕ ‖fj (n)‖ ≤ 2m∑

j=1

ϕ ‖fj (n)‖ .

Consequently g = (g(n)) ∈ lϕ(G). We claim that g is a best simultaneous approximationfor f1, f2, ..., fm ∈ l

ϕ(X) in l

ϕ(G). To see that let h ∈ l

ϕ(G). Then

m∑j=1

‖fj − h‖ϕ

=m∑

j=1

∞∑n=1

ϕ ‖fj (n)− h(n)‖

=∞∑

n=1

m∑j=1

ϕ ‖fj (n)− h(n)‖

≥∞∑

n=1

m∑j=1

ϕ ‖fj (n)− g(n)‖

=m∑

j=1

∞∑n=1

ϕ ‖fj (n)− g(n)‖

=m∑

j=1

‖fj − g‖ϕ.

�

References

[1] I. Abu-Sarhan and R. Khalil, Best simultaneous approximation in vector valued function spaces.Int. Journal of Mathematical Analysis 2(2008) 207-212.

80 SH. AL-SHARIF

[2] L. Chong and G.A. Watson, On the best simultaneously approximation, J. Approx. Theory 91(1997)332-348.

[3] W. Deeb and R. Khalil, On the tensor product of non-locally convex topological vector spaces,Illinois J. Math. Proc. 30 (1986), 594- 601.

[4] W. Deeb and R. Khalil, best approximation in Lp(I,X), 0 < p < 1 J. Approx. Theory 58 (1989),68-77.

[5] W. Deeb and R. Younis, Extreme points in a class of non-locally convex topological vector spaces,Math. Reb. Toyama Univ. 6(1983),95-103.

[6] R. Khalil and W. Deeb, best approximation in Lp(I, X), ii J. Approx. Theory 59(1989), 296-299.[7] R. Khalil and F. Saidi, best approximation in L1(I,X), Proc. Amer. Math. Soc. 123(1995), 183-190.[8] W. Light, Proximinality in Lp(S, Y ), Rocky Mountain J. Math. (1989).[9] J. Mach, Best simultaneous approximation of vector valued functions with values in certain Banach

spaces, Math. Ann. 240(1979), 157-164.[10] J. Mendoza, Proximinality in Lp(µ,X), J. Approx. Theory, 93 (1998), 331-343.[11] P.D.Milman, On best simultaneous approximation in normed linear spaces, J. Approx. Theory

20(1977), 223-238.[12] A. Pinkus, Uniqueness in vector valued approxomation, J. Approx. Theory 73(1993), 17-92.[13] B.N.Sahney and S.P.Singh, On Best simultaneous approximation in Banach spaces, J. Approx.

Theory 35(1982), 222-224.[14] F. Saidi, D. Hussein and R. Khalil, Best simultaneous approximation in Lp(I, X), J. Approx.

Theory 116(2002), 369-379.[15] S. Tanimoto, on of best simultaneous approximation, Math. Japonica 48(1998), 275-279.[16] G.A. Watson,, A characterization of best simultaneous approximation, J. Approx. Theory

75(1993),175-182.

Department of Mathematics, Yarmouk University,Irbid JordanE-mail address: [email protected]

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ALGEBRAIC MODELS IN APPLIED RESEARCH

THOMAS VOUGIOUKLIS and PENELOPE KAMBAKI

ABSTRACT:

Mathematical models can be used in several sciences as a tool to organize the research in such a way that the results could be transferred into other sciences. Here we go on with a method how to organise some results, which do not seem to have any relation with mathematics in a strictly algebraic way. The topic of algebra used is called Hv-structures, the largest class of hyperstructures. Our examples are taken from recent research in linguistics, more specifically on sociolinguistics. We believe that this game with mathematics will stimulate the interest of both researchers and students; moreover it is within the frame of Interdisciplinary Approach, one of the most recent teaching attitudes.

AMS Subject Classification: 20N20. Key words: hyperstructures, Hv-structures.

1. INTRODUCTION

In this paper we get on with our proposed method for modelling in research by using the special algebraic domains called hyperstructures. This method is motivated by research in linguistics and this is where we apply this modelling. Our point is that, during the research and teaching process, it is interesting to see if the results could be possibly formulated in an algebraic domain. Such a process could be applied in many ways bringing to light interesting aspects connecting pure research with applications in the classroom environment.

We deal with hyperstructures called Hv-structures introduced by T. Vougiouklis, in 1990 [5], which satisfy the weak axioms where the non-empty intersection replaces the equality. This topic is growing rapidly as one can see in the site: aha.eled.duth.gr.

2. Hv-STRUCTURES

We recall some basic definitions mainly from [6]:

In a set H equipped with a hyperoperation ⋅:H×H→P(H)-{∅}, we abbreviate by

WASS the weak associativity: (xy)z∩x(yz) ≠ ∅, ∀x,y,z∈H and by

COW the weak commutativity: xy∩yx ≠ ∅, ∀x,y∈H.

The hyperstructure (H,⋅) is called Hv-semigroup if it is WASS and it is called Hv-group if it is reproductive Hv-semigroup, i.e. xH = Hx = H, ∀x∈H.

The hyperstructure (R,+,⋅) is called Hv-ring if (+) and (⋅) are WASS, the reproduction axiom is valid for (+) and (⋅) is weak distributive with respect to (+):

x(y+z)∩(xy+xz) ≠ ∅, (x+y)z∩(xz+yz) ≠ ∅, ∀x,y,z∈R. Received: Sept 5,2007 Accepted: Jan. 3,2008

Jordan Journal of Mathematics and Statistics ( JJMS) 2008, 1(1) , pp. 81 - 90

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For more definitions such as Hv-vector spaces, Hv-algebras or Hv-Lie algebras and applications on Hv-structures, see the books [6],[2],[3], review papers as [4] and related papers as [1], [7].

The fundamental relations β*, γ* and ε* are defined, in Hv-groups, Hv-rings and Hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector space, respectively [6]. A way to find the fundamental classes is given by analogous theorems to the following [5],[6],[7]:

Theorem 1. Let (H,⋅) be a Hv-group and denote by U the set of all finite products of elements of H. We define the relation β in H by xβy iff {x,y}⊂u where u∈U. Then the fundamental relation β* is the transitive closure of β.

An element is called single if its fundamental class is singleton.

The fundamental relations are used for general definitions. Thus, in order to define the Hv-field the γ* is used: A Hv-ring (R,+,⋅) is called Hv-field if R/γ* is a field. In the sequence the Hv-vector space is defined [7],[10].

Let (H,⋅), (H,*) be Hv-semigroups. (⋅) is called smaller than (*), and (*) greater than (⋅), iff there exists an

f∈Aut(H,*) such that xy⊂f(x*y), ∀x,y∈H.

Then we write ⋅≤* and we say that (H,*) contains (H,⋅). If (H,⋅) is a structure then it is called basic structure and (H,*) is called Hb-structure.

Theorem 2 [6]. Greater hyperoperations of the ones which are WASS or COW are also WASS or COW respectively.

Definitions [8],[9],[10]. Let (H,⋅) be a hypergroupoid. We remove h∈H, if we take the restriction of (⋅) in H-{h}. h∈H absorbs h∈H if it replaces h so h does not appear in the structure. h∈H merges with h∈H, if the product of any x∈H by h, is the union of the results of x with both h, h, and consider h and h as one class with representative h.

Therefore one can add or remove elements in Hv-structures and the obtained Hv-structures have new and old properties.

The Hv-structures are used in Representation Theory. Representations of Hv-groups can be considered either by generalized permutations or by Hv-matrices [6]. The representations by generalized permutations can be achieved by using left or right translations. The single elements are playing a crucial role.

Definitions [6]. Let (H,⋅) be Hv-group, then the powers of an element x∈H, using the circle hyperoperation (⊗), i.e. take the union of all hyperproducts putting the parentheses on all possible ways, are defined as follows:

x1 ={x}, x2 = x⊗x, . . . , xs = x⊗…⊗x, . . .

The Hv-group (H,⋅) is called cyclic with finite period s, the minimum one, with respect to the generator h∈H, if

H = h1∪ h2∪ . . . ∪hs

If all generators have the same period, then H is cyclic with period. If there exists h∈H and s, the minimum one, such that H = hs, then H is a single-power cyclic, h is a generator with single-power period s. There is no any analogous definition in the classical theory, moreover one can define the infinite cyclicity as well.

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3. THETA HYPERSTRUCTURES

In our modelling we need the following: In [11] a hyperoperation in a groupoid with a map f on it, is defined, which is denoted by theta ∂.

Definitions 3. Let H be a set equipped with n operations (or hyperoperations) ⊗1,⊗2,…,⊗n and a map (or multivalued) f:H→H, then n hyperoperations ∂1,∂2,…,∂n on H can be defined, called theta-operations by putting

x∂iy = {f(x)⊗iy, x⊗if(y) }, ∀x,y∈H and i∈{1,2,…,n}

or in case where ⊗i is hyperoperation or f is multivalued map we have

x∂iy = ( f(x)⊗iy)∪(x⊗if(y) ) , ∀x,y∈H and i∈{1,2,…,n}

If ⊗i is associative then ∂i is WASS.

Let (G,⋅) be groupoid, fi:G→G, i∈I, be a set of maps, then the union of the fi(x) is

f∪:G→ P(G) such that f∪(x)={fi(x)⏐i∈I }

The union theta-operation (∂) on G is obtained if we consider the f∪(x). A special case is the union of a map with the identity: f ≡f∪(id), so f(x)={x,f(x)}, ∀x∈G, is called b-theta-operation. We denote by (∂) the b-theta-operation, so we have

x∂y = { xy, f(x)⋅y, x⋅f(y) }, ∀x,y∈G.

Motivation for the definition of the theta-operation is the map derivative where only the multiplication of functions can be used. Therefore, in these terms, for two functions s(x), t(x), we have s∂t={s′t, st′} where (′) denotes the derivative.

Properties 4 [11]. If (G,⋅) is a semigroup then:

(a) For every f, the hyperoperation (∂) is WASS.

(b) For every f, the b-theta-operation (∂) is WASS.

(c) If f is homomorphism and projection, i.e. f2 =f, then (∂) is associative.

Proof. (a) For all x,y,z in G we have

(x∂y)∂z = { f(f(x)⋅y)⋅z, f(x)⋅y⋅f(z), f(x⋅f(y))⋅z, x⋅f(y)⋅f(z) }

x∂(y∂z) = { f(x)⋅f(y)⋅z, x⋅f(f(y)⋅z), f(x)⋅y⋅f(z), x⋅f(y⋅f(z)) }

Therefore (x∂y)∂z∩x∂(y∂z) = { f(x)⋅y⋅f(z) }≠∅, so (∂) is WASS.

(b) and (c) are proved similarly. ■

Properties 5. Reproductivity. If (⋅) is reproductive then (∂) is also reproductive:

x∂G = ∪g∈G{f(x)⋅g, x⋅f(g)}= ∪g∈G{f(x)⋅g}= G,

G∂x = ∪g∈G{f(g)⋅x, g⋅f(x)}= ∪g∈G{g⋅f(x)}= G

Commutativity. If (⋅) is commutative then (∂) is commutative, if (⋅) is COW then (∂) is COW.

Unit elements: u is a right unit element if x∂u={f(x)⋅u, x⋅f(u)}∋x. So f(u)=e, where e is a unit in (G,⋅). The elements of the kernel of f, are the units of (G,∂).

Inverse elements: let (G,⋅) is a monoid with unit e and u be a unit in (G,∂), then f(u)=e. For given x, the element x′ is an inverse with respect to u, if

x∂x′= {f(x)⋅x′, x⋅f(x′)}∋u and x′∂x={f(x′)⋅x, x′⋅f(x)}∋u.

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So, x′ = (f(x))-1u and x′ = u(f(x))-1, are the right and left inverses, respectively. We have two-sided inverses iff f(x)u = uf(x).

Similar properties for multivalued maps are obtained.

Finally, since (∂) is WASS and the reproductivity is valid, we have the following

Proposition 6 [11]. Let (G,⋅) be group then, for all f: G→G, the (G,∂) is a Hv-group.

4. THE SOCIOLINGUISTIC EXPERIMENT

Now we present an experiment on linguistics on which we apply our modelling.

Introduction

The application of the suggested method of modelling was tested in a sociolinguistic experiment conducted in the Greek Department of Democritus University of Thrace, in spring, 2004. We are actually investigating a very interesting aspect of sociolinguistic research, namely the language of young people. More specifically, we want to find out which are some of the most popular and commonly used expressions among students, mainly of the Greek Department but also by other students of different departments, in Komotini [13], during the actual period of the experiment, as well.

Taking into account the fact that about 200 students – mainly female and few male- had to attend the obligatory subject of Introduction to General Linguistics during their first year of study, we should focus on two important points: first, all students were still unaware of linguistic research and what exactly or vaguely was sought in the specific experiment-consequently unsuspicious and ideal for this type of research; and second, there was the lure of a possible bonus, i.e. a rewarding higher score in the exams; and this is fair enough, as in any activity of this type there is always a form of reward to motivate the participants. This happy coincidence gave us the opportunity to choose amongst the most interested students and to create a group of willing to learn and conscious subjects who would also act as researchers themselves – although they did not know it at that point. The final selection was made on the basis of an expressed interest and motivation, followed by an introduction made by the tutor roughly explaining the methodology and pointing out the most important parameters. Fairly enough, many students were discouraged at this point and lost interest.

Method

Subjects

A total of 46 students finally participated, selected in the above described procedure and were allocated into nine groups – teams - for the needs of present study-of four members each and two teams of five, a total of eleven. All participating students were rewarded with an extra mark on a ten-range scale, provided they would have achieved a pass, i.e. a score higher than 5 out of 10. For example, if they would achieve a 5 out of 10, they would be given 6, if 7, they would be given 8, if 9 out of 10, they would be given 9, a fact that was highly appreciated. We call the subjects informants-researchers as they carry both these two properties in the specific piece of research.

Procedure, materials and tasks

The procedure and the tasks to be accomplished fall in the following four-step process:

─ 1st step: the teams scattered all over the amphitheatre so as to occupy a certain area well away from the other teams and privacy to be secured. Every single member, i.e.

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every informant-researcher, of each team had to decide all by her/himself about the 25 most commonly used expressions by young students of her/his environment and write them down in an order of frequency, starting with the items of the highest frequency, of course in her/his own, personal, subjective opinion.

─ 2nd step: after having completed their personal lists, the members of each team, still isolated from the other ten teams, met together, chose a secretary and wrote down a list of every expression each of them had recorded when working alone. The possibilities we could have here ranged from 25 -word lists, if everybody had written the same words, to 100-word lists, if every participants had written totally different words. Both cases were extreme and they did not finally occur, as one should expect.

─ 3rd step: still trying not to be overheard or overseen by the other teams, the members of each team have to decide whether the list they have come up with is really representative of the most commonly used expressions by young people of their environment, as well as if the order of frequency corresponds to the reality as they perceive it. This is a very demanding and time consuming process, as they have to conduct a lot of discussion and use arguments supporting their opinions, for example, why their personal classifications vary so much or why the divergence from the final classification is so great, etc. Needless to say, the role of the secretary becomes more demanding and consequently more important in the whole process; furthermore the teacher should go around and support the secretaries without interfering in the discussion, though.

─ 4th step: the final task of our informants-researchers was to find five different students from Komotini and ask them to name the five most commonly used expressions by young people. Extra attention should be paid so as that the informants our researchers had chosen had not participated in the experiment before and that they should answer as spontaneously as possible.

Discussion

We finally came up with eleven different classifications compiled by the eleven different teams. In order to exploit as many as possible examples, in present piece of research, we take the first five items from each classification. We do so because we feel that these specific items are the ones most likely to appear after a process of objectification. Hence we numbered these specific items and we have the following results:

t1(1,2,3,4,5), t2(6,7,3,8,9), t3(6,10,11,3,12), t4(6,13,14,15,16),

t5(17,6,13,4,16), t6(6,10,11,17,7), t7(6,18,16,19,20), t8(21,17,1,22,23),

t9(6,23,14,8,24), t10(6,25,26,27,28), t11(6,29,2,30,20),

where the numbers in parenthesis indicates the ordered expressions.

5. THE MATHEMATICAL MODEL

We present now some examples on our modelling using results of the above experiment from [14] in order to see the variety of the topic by using only theta operations.

Example 7. Let us reduce the number of teams in order to see the models with not a lot of calculations. We exclude teams t1, t5 and t8 because they do not have the most common expression no 6. Then we exclude t11 because it has no common expression with the other teams and, finally, we exclude teams t7 and t10 because they have less

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connection with the remaining. Therefore, we renumber the remaining teams into the model of Z5 as follows

1 ≡ t2, 2 ≡ t3, 3 ≡ t4, 4 ≡ t6, 0 ≡ t9.

So, if we exclude the most common expression we have the following:

0 (23,14,8,24), 1 (7,3,8,9), 2 (10,11,3,12), 3 (13,14,15,16), 4 (10,11,17,7).

Let us define the map f as follows:

f(i) = { j│the element (team) j ≠ i and has at least one common expression with i }

More precisely, in our example we have

f(0) = {1,3}, f(1) = {0,2,4}, f(2) = {1,4}, f(3) = {0}, f(4) = {1,2}.

Therefore, for example, we have

2∂3 = (f(2)+3 ) ∪ ( 2+f(3)) = ({1,4}+ 3 ) ∪ ( 2+{0}) = {2,4},

and so on. Taking into account that (+) is commutative then (∂) is also commutative we obtain the following table

∂ 0 1 2 3 4

0 1,3 0,2,4 0,1,3,4 0,1,4 0,1,2

1 0,2,4 0,1,3 0,1,2,4 0,1,2,3 1,2,3,4

2 0,1,3,4 0,1,2,4 1,3 2,4 0,3,4

3 0,1,4 0,1,2,3 2,4 3 0,4

4 0,1,2 1,2,3,4 0,3,4 0,4 0,1

From this table we obtain that the hyperstructure (Z5,∂) is an abelian Hv-group where the element 1 is a two-sited unit. The (Z5,∂) is cyclic where the elements 0,1,2 and 4 are generators with period 3. The (Z5,∂) is a single-power cyclic where the elements 0,1,2 and 4 are generators with period 4, 3, 4 and 3 respectively. Finally, the element 3 is an idempotent.

Example 8 [14]. In order to specify the teams under modelling one can use the experience of the researcher on the way the teams worked.

Different correspondence in Z8 gives other hyperstructures, sometimes more interesting in hyperstructure theory. For example, we can exclude the teams 1 and 8, since they do not include the most common expression and then the team 10 because this team do not have any common expressions with the others. The remaining 8 teams have the following four, except the common expression:

t2(7,3,8,9), t3(10,11,3,12), t4(13,14,15,16), t5(17,13,4,16),

t6(10,11,17,7), t7(18,16,19,20), t9(23,14,8,24), t11(29,2,30,20).

We define the following algorithm to obtain the map:

f(ti) = { tj │ tj ≠ ti and has the first expression of ti .

If this set is empty then we take the second expression of ti and so on}

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Therefore we have

f(t2)={t6}, f(t3)={t6}, f(t4)={t5}, f(t5)={t6},

f(t6)={ t3}, f(t7)={t4,t5}, f(t9)={t4}, f(t11)={t7}.

The theta operation (∂) is more interesting if there exists a unit element, we can select, for example, t6 to be the element 0 of (Z8,∂), because it appears as a singleton in the above f three times.

Finally, we renumber the teams in the model of Z8 as follows

0 ≡ t6, 1 ≡ t2, 2 ≡ t3, 3 ≡ t4, 4 ≡ t5, 5 ≡ t7, 6 ≡ t9, 7 ≡ t11.

Or in the new enumeration in Z8 we have

0 (10,11,17,7), 1 (7,3,8,9), 2 (10,11,3,12), 3 (13,14,15,16),

4 (17,13,4,16), 5 (18,16,19,20), 6 (23,14,8,24), 7 (29,2,30,20),

Therefore, we have the following:

f(0) = {2}, f(1) = {0}, f(2) = {0}, f(3) = {4},

f(4) = {0}, f(5) = {3, 4}, f(6) = {3}, f(7) = {5}.

and the following table is obtained

∂ 0 1 2 3 4 5 6 7

0 2 0,3 0,4 4,5 0,6 3,4,7 0,3 1,5

1 0,3 1 1,2 3,5 1,4 4,5 4,6 1,7

2 0,4 1,2 2 3,6 2,4 5,6 5,6 7

3 4,5 3,5 3,6 7 0,3 1,6,7 2,6 0,3

4 0,6 1,4 2,4 0,3 4 0,5,7 6,7 1,7

5 3,4,7 4,5 5,6 1,6,7 0,5,7 0,1 0,1,2 2,3

6 0,3 4,6 5,6 2,6 6,7 0,1,2 1 2,3

7 1,5 1,7 7 0,3 1,7 2,3 2,3 4

From this table we obtain that the hyperstructure (Z8,∂) is an abelian Hv-group where there are three unit elements, the elements 1, 2, and 4. We can obtain, after some calculations, that the (Z8,∂) is cyclic where the elements 0, 3, 5 and 6 are generators with period 8, 6, 4 and 7, respectively. The (Z8,∂) is a single-power cyclic where the elements 0, 3, 5 and 6 are generators with period 8, 6, 5 and 7, respectively.

6. THE SECOND EXPERIMENT

Two years later we applied the lists obtained from our experiment to a new group of informants. This time we chose academics, all doing teaching and research in the Department of Greek, aged ±40, mainly females and a few males. Their task was to put a tick next to each phrase of the list they themselves also used in production. The process was actually very quick and spontaneous in order to achieve a good degree of objectivity. They all enjoyed the test, some asked to take it with them to answer taking

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their time but they were not allowed to. Certain phrases are defined as ‘not used’ if the number of the choices is according to the Golden Ratio either in the normal ratio ≈0.618, or the compliment ≈0.382.

Example 9. For the second experiment using the Golden Ratio, we obtain that the expressions used by the young people but not by the older- still young enough- are the expressions 7,9,13,14,16,18,19,21,24,28. The teams which have at least one of the above expressions are

t2(7,9), t4(13,14,16), t5(13,16), t6(7), t7(16,18,19), t8(21), t9(14,24), t10(28).

The teams t8, t10 do not have any connection with the rest, therefore in (Z6,+) we have

0 ≡ t2(7,9), 1 ≡ t4(13,14,16), 2 ≡ t5(13,16), 3 ≡ t6(7), 4 ≡ t7(16,18,19), 5 ≡ t9(14,24).


f(i) = { j │the team j ≠ i and has at least one common expression with i }

Thus

f(0) = {3}, f(1) = {2,4,5}, f(2) = {1,4}, f(3) = {0}, f(4) = {1,2}, f(5) = {1}.

Therefore, for example, we have

2∂4 = (f(2)+4 ) ∪ ( 2+f(4)) = ({1,4}+ 4 ) ∪ ( 2+{1,2}) = {2,3,4,5},

and so on. Therefore, we obtain the following table

∂ 0 1 2 3 4 5

0 3 2,4,5 1,4,5 0 1,2 1,2

1 2,4,5 0,3,5 0,1,2,4,5 1,2,5 0,2,3 1,2,3,4

2 1,4,5 0,1,2,4,5 0,3 1,2,4 2,3,4,5 0,3

3 0 1,2,5 1,2,4 3 4,5 4,5

4 1,2 0,2,3 2,3,4,5 4,5 0,5 0,1,5

5 1,2 1,2,3,4 0,3 4,5 0,1,5 0

From the above table, after some calculations, we obtain the following:

(a) The element 3 is the only one unit element.

(b) The set {0,3} is a subgroup of the ∂–group (Z6,∂).

(c) The (Z6,∂) is a cyclic Hv-group where the elements 1, 2, 4, 5 are generators with period, respectively 3, 3, 4, 5 and single-power cyclic where the elements 1, 2, 4, 5 are single-power generators with period 4, 4, 4, 5, respectively.

Example 10. Using the second experiment again we consider the expressions according to the compliment of the golden ratio on the answers. The expressions used by the teams are

t1(non), t2(6,7,9), t3(6,10), t4(6,13,14,16), t5(13,16), t6(6,7,10),

t7(6,16,18,19,20), t8(21), t9(6,14,24), t10(6,28), t11(6,20).

Now, we do not consider the most common expression (6), and consequently teams t8 and t10 are excluded. Then we rename the teams as follows:

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0 ≡ t2(7,9), 1 ≡ t3(10), 2 ≡ t4(13,14,16), 3 ≡ t5(13,16),

4 ≡ t6(7,10), 5 ≡ t7(16,18,19,20), 6 ≡ t9(14,24), 7 ≡ t11(20).


f(i) = { j │the team j ≠ i and has at least one common expression with i }

Thus

f(0) = {4}, f(1) = {4}, f(2) = {3, 5,6}, f(3) = {2,5},

f(4) = {0,1}, f(5) = {2,3,7}, f(6) = {2}, f(7) = {5}.

Then we obtain the following table

∂ 0 1 2 3 4 5 6 7

0 4 4,5 3,5,6 2,5,7 0,1 1, 2,3,7 2 3,5

1 4,5 5 4,6,7 3,6,7 0,1,2 0, 1,3,4 2,3 3,6

2 3,5,6 4,6,7 0, 5,7 0, 1, 4,6,7 1, 2,3,7 0, 1, 2, 3,4,6 1,3,4 2,4,5,7

3 2,5,7 3,6,7 0, 1, 4,6,7 0,5 1, 3,4,6 2, 5,6,7 0,3,5 0,1,4

4 0,1 0,1,2 1, 2,3,7 1, 3,4,6 4,5 3,5,6,7 6,7 0,1,7

5 1, 2,3,7 0, 1,3,4 0, 1, 2, 3,4,6 0, 5,6,7 3, 5,6,7 0, 4,7 0, 1,5,7 1,2,6

6 2 2,3 1,3,4 0,3,5 6,7 0, 1,5,7 0 1,3

7 3,5 3,6 2,4,5,7 0,1,4 0,1,7 1,2,6 1,3 4

From the above table, we obtain the following:

(a) The element 4 is the only one unit element.

(b) The (Z8,∂) is a cyclic Hv-group where all its elements: 0, 1, 2, 3, 4, 5, 6, 7, are generators with period, respectively 6, 4, 3, 4, 5, 3, 5, 5 and single-power cyclic, except the element 0, with period 5, 3, 4, 5, 4, 7,6, respectively.

7. CONCLUSIONS

The above modelling provides a game between hyperstructures and applications, between Hv-structures and linguistics. This modelling gives on the one side new Hv-structures and on the other side it provides an organizing devise on applied research which is expected to result in boosting the interest of both the teachers and learners in the classroom environment.

On the algebraic point of view, on hyperstructures, we can see, for example that there are Hv-groups, which have unit elements, which are generators (see example 9).

It is of great interest, from the language teaching or linguistic research point of view, to examine whether the special elements such as the units, the generators or properties as associativity, cyclicity are due to the special action of the teams and the relations they have between them. Therefore, one can see why the elements 0 and 3 form a subgroup or what the special behaviour of the teams t2 and t6 is; nevertheless such an issue concerns mainly sociolinguistic research.

90

REFERENCES

[1] Ameri, R., Zahedi, M.M., Hyperalgebraic systems, Italian J. Pure Appl. Math. 6, 21-32, (1999).

[2] Corsini, P., Prolegomena of Hypergroup Theory, Aviani Editore, 1993.

[3] Corsini, P., Leoreanu, V., Applications of Hypergroup Theory, Kluwer Academic Publishers, 2003.

[4] Davvaz, B., A brief survey of the theory of Hv-structures, 8th AHA Congress, Spanides Press, 39-70, (2003).

[5] Vougiouklis, T., The fundamental relation in hyperrings. The general hyperfield, 4thAHA Congress, World Scientific, 203-211, (1991).

[6] Vougiouklis, T., Hyperstructures and their Representations, Monographs in Mathematics, Hadronic Press, 1994.

[7] Vougiouklis, T., Some remarks on hyperstructures, Contemporary Mathematics, American Mathematical Society, 184, , 427-431, (1995).

[8] Vougiouklis, T., Constructions of Hv-structures with desired fundamental structures, New frontiers in Hyperstructures, Hadronic Press,177-188, (1996).

[9] Vougiouklis, T., Enlarging Hv-structures, Algebras and Combinatorics, ICAC’97, Springer - Verlag, 455-463, (1999).

[10] Vougiouklis, T., On Hv-rings and Hv-representations, Discrete Mathematics, Elsevier, 208/209, 615-620, (1999).

[11] Vougiouklis, T., A hyperoperation defined on a groupoid equipped with a map, Ratio Mathematica on line, N.1, 25-36, (2005).

[12] Vougiouklis, T., The ∂ hyperoperation, Proc. Structure Elements of Hyperstructures, Alexandoupolis, Greece, 53-64, (2005).

[13] Vougiouklis, T., Kambakis P., On the Mathematics of the Language, Proc. 2nd Pan-Hellenic. ‘New technologies…’ Athens, Greece, 486-491,(2000).

[14] Vougiouklis, T., Kambakis P., Mathematical Models in Teaching and Research, Proc. 5th Mediterranean Conference 2007, Rhodes, Greece, 625-634,(2007).

THOMAS VOUGIOUKLIS, PENELOPE KAMBAKI Democritus University of Thrace Democritus University of Thrace School of Education Department of Greek 681 00 Alexandroupolis, Greece 681 00 Komotini, Greece [email protected] [email protected]

91

The Set of Values of Functionals in the Classes of Functions Having Integral Representation

Hassan Baddour

ABSTRACT: This paper presents a certain method to determine the set of values of the functional

)( fJ ( ) ( ) ( ) 0

zfza kn

kk∑

=

=

defined on the family of analytic functions in the unit disc possessing an integral representation with kernel ( )tz,q in the interval [ ]βα , . It has been shown that the set of values of the linear functional [ ])zf(J at the point 0z is the convex hull of the curve Γ, given by equation

( ) ( ) ( ) )t,z(qza tw : 0k

0

n

0=kk∑=Γ , βα ≤≤ t .

The Class of Functions with Limited Rotation was cosidered as a special case and some new results were obtained .

1. Introduction One of the fundamental extremal problems considered in the domain of complex functions is concerned with determining the range of variability of the functional

( ) ( ) ( ) ( )( )[ ]0n

00 zf,...,z' f,zf F =f J defined on some family E of analytic functions where 0z is a fixed point of the domain

in which the functions f are defined. A survey of methods and results can be found among others in [1] and [5].

In this paper we shall present a certain method to determine the range of variability (or

the set of values) of the linear functional ( )( )n , ... ,' , F =)( ffffJ ( ) ( )( ) 0

00 zfza k

n

kk∑

=

=

defined in the family of functions possessing an integral representation in the unit disk (the precise definition of will be given in the next paragraph).This method can be also applied to solve the coefficients problem in classes considered , i.e. if the function is given by Taylor Series then the coefficients can be strictly estimated in some special cases .

Jordan Journal of Mathematics and Statistics ( JJMS) 2008, 1(1) , pp. 91 - 96

Received: Dec. 12, 2007 Accepted: March 4 ,2008

92

2. The problem in the Class qE Let qE denote the class of functions f given by the formula

(1) ( ) ( )∫=β

αµ tz,)( tdqzf

where ( )tz,q is an analytic function in the unit disc ( )1z D < for every fixed [ ]βα ,∈t and )( tµ is a nondecreasing function in the interval [ ]βα , such that ( ) ( ) 1=− αµβµ . This class is known as the family of functions possessing an integral representation .It is to be proved that [6]: 1- The class qE is compact and connected in the topology of almost uniform convergence. 2- The set of values of functional )()( 0zffF = where qEf ∈ is closed , convex and connected and it is at the same time the convex hull of the curve ( ) ( ) βα ≤≤= ttzqtw ,, (The convex hull of the set W is the smallest convex set containing the set W ).

An essential example of this class is the Caratheodory Class ( denoted by C) i.e. the class of analytic functions f in the unit disk D with a positive real part and ( ) 10f = . It is well known ( [2] and [4]) that for the functions of this class the following integral representation holds:

(2) ( )tdzezezf it

it

µπ

π∫+

− −+

= )(

where ( )t µ in this case is the set of nondecreasing functions on the interval [ ]ππ ,− such that ( ) ( ) 1=−−+ πµπµ ). We see that the class C coincides with the family qE when we put

( ) ( ) ),( 00 zezetzq itit −+= and πα −= , πβ = . We shall make use of the following lemma:

Lemma1. If ( )tW is a complex and continuous function with a real variable t on the interval ],[ ba and ],[ baU the set of nondecreasing functions )( tµ on this interval such that , 0)( =aµ 1)( =bµ , then the set of values of the integral

( )∫=b

atd)t(W)(J µµ , [ ]baU ,∈µ

is the convex hull of the curve Γ given by equation . bta , )t(WW : ≤≤=Γ The proof of this lemma can be found in [4] . Making use of this lemma we shall prove the following: Theorem 1. If 0,1,...nk ,)z(ak = are some continuous functions on the unit disc D and

0z is some fixed point in this disc, then the set of values of the linear functional

(3) ( ) ( ) ( )( ) qk

n

kk EfzfzafJ ∈= ∑

=

, 00

0

93

is the convex hull of the curve Γ, given by equation

( ) ( ) ( ) )t,z(qza t,zW 0k

z0

n

0=kk0 ∑=

where )t,z(q 0 are given by (1) .

Proof. Differentation of both sides in the formula (1) gives the relations:

(4) ( ) ( )( ) ( ) .0,1,2,3,..=k tdtz,q )z(f kz

k ∫=β

αµ ,

provided that ( ) )z(f)z(f 0 = .And so

( ) ( ) ( ) ( ) ( ) ( ) ( ) .)t(dt,zqza )t(dt,zq zafJ kz

n

0kk

kz

n

0kk ∫ ∑∫∑

====

β

α

β

αµµ

Taking

(5) ( ) ( ) ( ) )t,z(qza t,zW 0k

0

n

0=kk0 ∑=

we obtain the functional (3) depends only on )( tµ and so we can write

( ) ( ) ( ).tdt,zW J 0∫==β

αµµΦ

Substituting βα == b ,a and )t,z(W)t(W 0= in Lemma 1 we get the thesis of theorem.

3. The problem in the class of functions with limited rotation.

Let V denote the family of analytic functions f in the unit disk D satisfying the conditions:

( ) ( ) ( ) 2z'farg ,10'f ,00f π≤== This family is known as the class of functions with limited rotation. Lemma 2. If Vf ∈ then : a) Cf ∈' . b) qEf ∈

Proof. a) If Vf ∈ then ( by the definition) ( ) 10'f = . On the other side the condition ( ) 2'arg2 ππ <<− zf implies that

( ) ( ) ( ) 0z'fargcos'fe'frez'f re z'fargi >==

and hence Cf ∈' .

b) From the fact that Cf ∈' and formula (2) we conclude that

( ) dz]td zeze [)z(f it

itz

0µ

π

π∫∫+

− −+

=

And hence the functions of V has the following integral representation:

94

( ) ( )td]dz zeze [zf it

itz

0µ

π

π∫ ∫+

− −+

=

In this way we see that qEf ∈ with πβπα =−= , , and

( ) dz zezet,zq it

itz

0 −+

= ∫ .

Theorem 2 . The set of values of the functional

(6) ( ) ( ) ( )( ) Vf ,zf za fJ 0k

n

0k0k ∈= ∑

=

is the convex hull of the curve Γ given by the equation

(7) ( ) ( ) ( ) ( )( )kit

it

0

n

2=kkit

it

01it

itz

0000

ze

e)!1k(2zazezeza

zezezat,zW

0

−

−+

−+

+−+

= ∑∫

where ππ ≤≤ t- . Proof. Notice that the equations (4) have , in this case, the form

( )( ) ( )zfzf 0 = , ( )td zeze )z('f it

itµ

π

π∫+

− −+

=

( ) ( )( )

( ) ,...3,2 , !12 )(k =−

−= ∫

+

−

ktdze

ekzf kit

it

µπ

π

and thus putting

( )

( )

( ) ( )( )

nkze

ektzq

zezetzq

dzzezetzq

kit

itk

it

it

it

itz

,...,3,2,)!1(2,

,,'

,,

0

0

00

=−

−=

−+

=

−+

= ∫

and substituting in (5) we get the curve Γ in the form (7) .And so the convex hull of this curve is the set of values of the functional (6). This ends the proof.

95

4. Applications. In what follows we will give some propositions as an application of the above theorem .

Proposition 1. The sets of values of functional )z(f)f(F 0= with Vf ∈ and Dz0 ∈ is the closed set G which is connected , symmetric with respect to the real axis

and contained in the disc d2zw 0 ≤− where ςς Gmaxd∈

= and ( )0it ze1Log −−=ς .

Proof. The curve Γ in this case has the form

( ) ππΓ +≤≤−+

= ∫ t- ,dzzezet,zW : it

itz

00

0

Putting ( ) ( )t,zWtW 0= and integrating with respect to z we get

(8) ( ) ( )0itit

0 ze1Loge2ztW −−−= .

Its known that [4] the principle branch of logarithm ( )z1Log −=ς maps conformally the closed disc 1r ,rz <≤ into the connected domain G which is closed and symmetric with respect to the real axis . If δ is the maximum of the magnitude of ς when ς varies on G (e.i. ς

ς Gmaxd∈

= ) then the set of values of ( )z1Log −=ς exists in

the disc d≤ς . This implies that the set of values of ( )0itit ze1Loge2 −−− exists in the

disc d2≤ς and hence the convex hull of the curve (8) would be contained the closed

disc d2zW 0 ≤− where ςς Gmaxd∈

= and ( )0it ze1Log −−=ς .

Proposition 2 The sets of values of functional

( ) ( ) ( ) ( ) ( ) zfz'fzffJ n++= Vf ∈

at the point 0z0 = is the closed disc with center at 1 and rarius ( )!1n2 − . Proof. Since

( ) ( ) ( ) ( ) 1,-n2,3,...,k ,0za ,1zazaza 0kn10 ===== And 0z0 = in theorem 2 , then the equation of Γ in this case has the form

( ) tW = ( ) ( ) ( )( )t,0q t,0'q t,0q n++ =( )nit

itz

0 e

e)!1n(21dz0 −

++∫

which implies that Γ is the circle ( ) ππ ≤<−+= +− t- ,e)!1n(21)t(W t1ni

And so the set of values of the given functional is the convex hull of Γ which represents in this case the closed disc ( )!121 −≤− nW .

Now, let Vf ∈ and let

(9) ( ) 1z ,...za...zazzf nn

22 <++++=

96

be expantion of f in Taylor Series . It is well known [6] that the coefficients na in this case satisfy the estimates n2a n ≤ for every n . We will show this fact using the above theorems.

Proposition 3. If (9) is the Taylor Expantion of Vf ∈ in the neighborhood of 0 then the following estimates hold:

... ,3 ,2 ,1n , n2a n =≤ Proof. Let us first find the set of values of the functional

( ) Vf ,)z(f)f(F n ∈= at the point 0=z . Noticing that the curve Γ in this case is given by the equation ( )( ) ( ) ( ) ππ +≤≤−= −− t- ,e!1n2t,0q =t)(0,W=W(t) t1nin (which represents a circle ( )!1n2W −= ) we can conclude that the closed disc

( )!1n2W −≤ is the convex hull of Γ and it is at the same time the set of values of the given functional . Now if (9) is the expantion of the function ( )zf in Taylor Series then

( )n

n a!n)0(f = .

On the other hand we have shown that the value ( ) )0(f n lies in the disc ( )!1n2W −≤ for every Vf ∈ . This implies the relation

( )!1n2a!n)0(f n)n( −≤=

which shows that the estimates ( )

n2

!n!1n2an =

−≤ hold for every n .

References

[1]. I. Aleksandrov , Boundary Values of Functional on the Class of Holomorphic Functions Univalent in a Circle . Sibirsk , Mat. Z. 4 , (1963) , 17-31.

[2] H. Baddour, About the range of variability of linear functionals in Caratheodory Classe. Damascus univ.journal- No.28 – 1998

[3] H. Baddour, Extremal problems in families of functions possessing a structural representation with the aid of measurable functions, Damasucs Univ.Journal,V16,No 1-2,2000.

[4] J. Krzyz , Theory and Problems in Analytic Functions. P.W.N Warsaw 1975 . [5] Ch. Pommerenke , Univalent Functions . Vandehhoeck & Ruprecht in

Go`ttingen 1975. [6] W. Rogosinski, Uber positive harmonishe Entwicklungen und typisch-

reelle Potenzreihen, Mat. Z. 35 (1932) p 93-121.

Department of Mathematics University of Tishreen Email: [email protected]

للرياضيات واإلحصاء

املجلة األردنية

مجلة بحوث علمية عاملية محكمة

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هـ1429 ربيع الثاني/ م2008 نيســـان، )1(العدد ، )1(المجلد ،ردن التعليم العالي والبحث العلمي، األوزارةللبحث العلمي، أسستها اللجنة العليا مجلة علمية عالمية محكمة: للرياضيات واإلحصاءاملجلة األردنية

.وتصدر عن عمادة البحث العلمي والدراسات العليا، جامعة اليرموك، إربد، األردن

: رئيس التحرير مشهور عبد الله الرفاعي .د.أ

قسم الرياضيات، جامعة اليرموك، اربد، األردن نائب رئيس جامعة اليرموك: الحاليالعنوان

E-mail: [email protected]

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مفيد محمد عزام. د.أ ردنية، عمان، األردنقسم الرياضيات، الجامعة األ


أحمد ذيب عالونة. د.أ قسم الرياضيات، الجامعة األردنية، عمان، األردن


بسام يوسف الناشف. د.أ ، األردنجامعة اليرموك، اربدقسم الرياضيات،


فؤاد أسعد كتانة. د.أ قسم الرياضيات، الجامعة األردنية، عمان، األردن


عبد الله محمد الجراح. د.أ قسم الرياضيات، جامعة اليرموك، اربد، األردن


.قسم الرياضيات، جامعة اليرموك، عبد الله الجراح. د.أ : العلمياملحرر

-: ان التالي البحوث إلى العنو ترسل األستاذ الدكتور مشهور عبد الله الرفاعي

المجلة األردنية للرياضيات واإلحصاءررئيس تحري األردن-اربد ، ة البحث العلمي والدراسات العليا، جامعة اليرموكعماد

2026 فرعي 00 962 2 7211111 هاتفE-mail: [email protected]

Website: http://jjms.yu.edu.jo

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