the essential spectral radius of dominated positive operators

The Essential Spectral Radius of Dominated Positive OperatorsAuthor(s): Josep MartínezSource: Proceedings of the American Mathematical Society, Vol. 118, No. 2 (Jun., 1993), pp.419-426Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2160318 .

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 118, Number 2, June 1993

THE ESSENTIAL SPECTRAL RADIUS OF DOMINATED POSITIVE OPERATORS

JOSEP MARTINEZ

(Communicated by Palle E. T. Jorgensen)

ABSTRACT. Let E be an ALp-space with 1 < p < oo. We prove that if a positive operator S e 5(E) satisfies the Doeblin conditions and r(S) < 1, then S is quasi-compact, i.e., ress(S) < 1 . We then deduce the following result about the monotonicity of the essential spectral radius: Let S, T e &(E) be such that 0 < S < T. If r(S) < 1 and ress(T) < 1, then ress(S) < 1.

0. INTRODUCTION

An interesting problem in the theory of positive operators in Banach lattices is to know what properties of T E 5f(E), where E is a Banach lattice, are inherited by S E B(E), if we know that 0 < S < T. In 1979 Dodds and Fremlin [4] started the investigations in this direction by studying the behaviour of compactness under domination.

Dodds-Fremlin Theorem. Let S, T: E -+ F be linear operators between two Banach lattices such that E' and F have order continuous norms. If 0 < S < T and T is compact, then S is compact.

Compactness properties have also been considered in [1] and related spectral properties in [3, 5, 6]. In particular, it has been studied if ress(S) < ress(T) when 0 < S < T. The monotonicity of the essential spectral radius is known in only a few cases: when S is an AM-compact operator (see [2, 8]) and when E is an AM-space or an AL-space (see [6, 10]).

In [3] Caselles proved the following general result closely related to the monotonicity of the essential spectral radius:

Theorem 0.1. Let E be a Banach lattice. Let S, T E Y(E) be such that 0<S< T. If r(T) <1 and ress(T) <1, then r,,,(S) <1.

Using the Doeblin condition, a similar result was stated in [5]:

Theorem 0.2. Let E = YP(yu) with 1 < p < oc and let S, T E 2 (E) be such that 0 < S < T. Assume that ress(T) < 1 and that the means Sn = Zn-1 Si are uniformly bounded. Then ress(S) < 1.

Received by the editors April 23, 1991 and, in revised form, September 21, 1991. 1991 Mathematics Subject Classification. Primary 47B55, 47A53. The author was supported by a F.P.I. grant from Spanish Ministerio de Educaci6n y Ciencia.

(D 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

419



420 JOSEP MARTINEZ

In this paper we prove Theorem 0.2 substituting the assumption sup{f ISn }I < 00 by the weaker one r(S) < 1, as we conjectured in [6]. Thus our result generalizes Theorems 0.1 and 0.2.

1. PRELIMINARIES

Throughout this paper, E will denote a complex Banach lattice, i.e., E =

ER e IER, the complexification of a real Banach lattice ER. The absolute value in ER is extended to E by means of the formula Ifl = sup{(cos 6)g + (sin 6)h: 0 < 0 < 27r}, where f = g + ih with g, h E ER (see [9, Chapter II, 11] or [I1, ?91]) and the norm in E satisfies llzlz = 11 lzl 11 for all z E E. Given x E E+, by the complex order interval [-x, x] we mean the subset [-x, x] = {z E E

ZIz 'X}- Given A C E we say that A is order bounded if A is included in some

complex order interval. A is called almost order bounded if for every e > 0 there exists u > 0 such that A c [-u, u] + 8BE, BE being the closed unit ball of E.

Let E be a complex Banach lattice. By 2 (E) we denote the space of all bounded linear operators from E into E. If T E Y(ER) then T has a unique extension as a linear operator on E by defining

T(f+ig)=Tf+iTg Vf,gEER.

Hence the set Y(ER) can be regarded as a real linear subspace of Y(E). Any operator in this subspace is called real. For T E Y(E) there exist canonical real operators Re T and Im T such that Tz = (Re T)z + i(Im T)z for all z E E. An operator T E Y(E) is said to be positive (T > 0) if T is equal -to ReT and Tz > 0 for z eE+.

If E is a complex Banach space and T E Y(E), we denote, as usual, the spectrum of T by v(T) and the spectral radius by r(T). Recall that T E Y(E) is called upper (lower) semi-Fredholm if T(E) is closed and its kernel (cokernel) Ker T(E/T(E)) is finite dimensional. The set of all upper (lower) semi-Fredholm operators will be denoted by ??+(E) (respectively, ??_(E)). Df(E) = @D+(E) n &_(E) is the set of the so-called Fredholm operators. The Fredholm domain of T E Y(E) is defined by

?(T) = { E C: (I- T) E @(E)}.

The set C - ?(DT) will be called the (Wolf) essential spectrum of T and will be denoted by aes (T). The essential spectrum of T is equal to the spectrum of the canonical image of T in the Calkin algebra Y(E)/I(E), where X(E) denotes the ideal of compact operators in E. The essential spectral radius is defined by

ress(T) = sup{|lA: A E aess(T)}.

We have that

ress(T) = lim(JJTn11ess)1/', where JJTJ1eSS=dist(T, X(E)). n

Finally, our general references concerning the theory of positive operators in Banach lattices are the monographs [1, 9, 1 1]. For the basic results on Co- semigroups and further information, see [7].



ESSENTIAL SPECTRAL RADIUS OF DOMINATED POSITIVE OPERATORS 421

2. THE ESSENTIAL SPECTRAL RADIUS AND THE DOEBLIN CONDITION

The Doeblin conditions are the main tool that we use in order to study the monotonicity of the essential spectral radius. The following definitions are taken from [5].

Definition 2.1. Let T be a positive operator on the Banach lattice E. (1) T is said to satisfy the Doeblin condition (which we note T E (D)) if

there exist m E N, 0 < u E E', and a real number q < 1 such that

II Tmxll < ,(x) + qllxll Vx E E+. (2) T is said to satisfy the dual Doeblin condition (which we denote by

T E (D')) if there exist m E N, 0 < x E E, and a real number q < 1 such that

TmBE C [-x, x]+ iBE,

where BE denotes the unit ball of E.

To prove our main result we need the following basic lemma. We omit its proof because it is a slight modification of Lemma V.5.3 in [9]. Lemma 2.2. Let E be a Dedekind complete Banach lattice. Let S E Y(E) and let J be a band such that S(J) c J. Denote by PJ1 the band projection onto J' . If re,,(S/J) < 1 and re,,(PjiS/J') < 1, then re,,(S) < 1.

It is well known that if T is a positive linear operator on the Banach lattice E then the spectral radius r(T) belongs to the spectrum a(T). However, in general, r(T) is not an eigenvalue of T. In the next result, which is based on Proposition 2 in [5], we state the existence of a positive eigenvector when T satisfies the Doeblin conditions.

Theorem 2.3. Let E be a Banach lattice and let S be a positive operator on E such that S E (D) n (D'). If r(S) = 1 then there exists a E E+, a :$ 0 such that Sa = a. Proof. Since S > 0 and r(S) = 1, there exists a normalized sequence (xn) of positive elements in E with limn 11(I - S)xn II = 0. (Take, for example, Xn = R(An, S)x/llR(n , S)xjj where x > 0, n -+ 1+, IIR(An, S)xI -+ oo .)

Let 1??(E) denote the Banach space of all bounded sequences (Yn) in E equipped with the supremum norm jj(Yn)lloj = Sup{jlynll n E N}, and let s(E) be the closed subspace of those sequences (Yn) in E such that their range {Yn: n E N} is almost order bounded. We denote by E the quotient Banach space 1??(E)/s(E). Since S is positive, S induces an operator S on E given by

S((Yn) +s(E)) = (SYn) +s(E) V(Yn) E l0(E)

(see [2]). As S E (D'), we get r(S) < 1. Now, since ((I - S)xn) E s(E), it follows that (xn) E s(E).

Now let A be the closed ideal generated by E in E". As (xn) E s(E), given e > 0, there exists u > 0 such that {xn} C [-u, u] + eBE . It follows that

{Xn} a(E",E') C [-u, u] + eBE a(E ,E' c u l + eBE",

where if-u, uj = {z e E" :Izi < u}. So we get {xn}a(E"E)CA. Now,as {xn} a(E" ,E) is a cT(E", E')-compact, there exist a subnet {Xnk} and a E A such that aT(E", E') - limk xnk = a, and so we obtain that S"a = a, a > 0.



422 JOSEP MARTINEZ

Since A c (E/)n (see [1, ?5]), E is oli(E", E') dense in A (see [1, 11.16]). So there exists {yy} C E such that {yy} c [0, a] and a(E", E')-limy Iyy-aI = 0. The Doeblin condition implies that, given e > 0, there exist ,u E E', no E N such that

ISnOyy -Sn?oy II < 1u(lyy -YI) + cIIyy -Yf II.

Given q E BE' , we obtain

l(Snoyy - Snoyf)I < 1u(lyy - al) + u(Iyf - al) + cHIyy - YgII.

It follows that there exists yo such that if y, ,B > yo then

I ((Sno)Iq$)yy - ?(Snfoyf) I < e + 2e sup IIYY I.

Taking limits for y, we get

ja((Sno)yk) - ?(Snoyfi)I <? + 2e sup IIyYII, y

and so

11(Sno)IIa -Snfoyflhl < e + 2c sup IIYY 11 y

Therefore we can write (Sno)"a = a E E and, consequently, a(E, E') - limxnk = a.

Since Sa = a, we only need to prove that a :$ 0. Suppose, on the contrary, that a = 0. Then the Doeblin condition implies that there exist m E N and a real number q < 1 such that lim supk IISmxnk < q < 1. On the other hand, since limn II(I - S)xnII = 0, we get that limn 11(I - SM)xn I = 0, hence

lim lSmxnk II = lim Ixnk|| = 1.- k k

This contradiction concludes the proof.

Remark 2.4. (a) If E is a reflexive Banach lattice then Theorem 2.3 also holds when S satisfies only the dual Doeblin condition (see [5, Proposition 3]).

(b) If E is an AL-space (resp. an AM-space), then each operator satisfies the Doeblin condition (resp. the dual Doeblin condition).

Suppose 1 < p < oCX. A Banach lattice E is called an abstract 5P-space (or an ALp-space) if

IIx+YII = (IxI +IIYIIP)'1P Vx,y EE+:xAy =O.

If 1 < p < 00 then any ALp-space is reflexive. Of course, any concrete YP(,L)-space is an abstract YP-space and it can be proved that the converse is also true (see [6, II, Example 23]). Obviously, each sublattice of an abstract YP-space is an abstract YP-space.

Our main result shows, roughly speaking, that a positive operator S in an abstract YP-space satisfying the Doeblin conditions is quasi-compact, i.e., re,,(S) < 1. First we prove a simplified version making use of the so-called ultrapower E of the Banach lattice E with respect to a free ultrafilter ?/ on N. For the construction of E, we refer to [9, V, 1.4].




Lemma 2.5. Let E be an abstract YP-space with 1 < p < oc. Let 0 < S E Y(E) such that S satisfies the dual Doeblin condition. If there exists a quasi- interior point x in E+ such that Sx = x and r(S) = 1, then re,,(S) < 1 .

Proof. As S E (D'), there exist no E N, y E E+ such that

Sn?BE C [-y, y]+ 2BE-

Since, by assumption, y E EX, we obtain that there exists m E N such that

Sn0BE C m[-x, x]+ 4BE-

If we define now z = m EZ 0(3)nx, from the above expression we get

SnonBE C [-Z, z] + (3)nBE.

Denoting by S the canonical extension of S to the ultrapower E, we have

SnonB.c C[-Z,r z 4 E

where z denotes the class of (z, z, z, . . .Consequently, if I1j = 1 then

(1) Ker() -S) nB?C[-z- ].

We recall that E2, the principal ideal generated by z, is an AM-space if we take {x' E E : Ix'l ? z} as its unit ball. Note that in this case we have the following factorization:

Ker() - S) L Ei2 - E

where il, i2 are the canonical inclusions. i2 is by definition a continuous map and it follows from (1) that il is a well-defined continuous operator.

(a) Suppose that E is an ALp-space with 1 < p < oX. Since E is an ALP- space, the reflexivity of E implies that il, i2 are weakly compact operators. Since E2 is a Dunford-Pettis space, i2 is a Dunford-Pettis map (see [ 1, Chapter 5, ? 19]). We conclude that i2 o il is a compact operator, whence Ker(A - S) is finite dimensional.

(b) Suppose that E is an AL-space. Then, as E2 is an AM-space and E is an AL-space, i2 o il is an integral map (see [9, IV, Theorem 5.4]). Moreover, S E (D') implies that each bounded sequence in Ker(A - S) is almost order bounded. As E has order continuous norm, each bounded sequence in Ker(A - S) is weakly compact. Therefore Ker(A - S) is reflexive. It follows that i2 o il is compact (see [9, IV, Theorem 5.4, Corollary 1 and Theorem 5.5, Corollary 2]). Therefore Ker(A - S) is finite dimensional.

Theorem 3.4 in [3] implies now that A V aess(S) whence ress(S) < 1

We now establish our main result:

Theorem 2.6. Let E be an ALp-space with 1 < p < oc. Let S be a positive operator on E such that r(S) < 1. If S satisfies the dual Doeblin condition then ress(S) < 1. Proof. Since ress(S) < r(S), we can assume that r(S) = 1. Then Remark 2.4 shows that there exists a positive element a, :$ 0 in E such that Sal = a, . If Eai denotes the principal ideal generated by a, then Eai is a closed ideal (in fact, Ea1 is a band because E has order continuous norm) with



424 JOSEP MARTINEZ

quasi-interior points. It is easy to see that S/Eai E (D'), and then it follows from Lemma 2.5 that ress(S/Eai) < 1. Now let us consider the operator (P(Ea )IS)/(Eai)I E ((Ea)'1) , where P(Ea) ' denotes the band projection onto

Ea-L * If r((P(Eal)?S)/(Eai)') < 1 then Lemma 2.2 implies ress(S) < 1, which concludes the proof.

Assume on the contrary that

r((P(Ea ) ? S)/Eal)) = 1.

Since (P(Ea )l S)/(Ea,)' E (D') , we get from Remark 2.4 that there exists a2 > 0

such that (P(Eal)tS)/(Ea,)'a2 = a2. Denote by Ea2 the closed ideal (band) generated by a2 . We can write

Ea2 = a2 S(Ea iE Ea2) CEal iE Ea2-

Lemma 2.5 implies that ress(Pa S/Ea2) < 1 and, as Ea2 is the band orthog-

onal to Eal in the Banach lattice Eal (3 Ea2, it follows from Lemma 2.2 that ress(S/Eai E Ea2) < 1. As a consequence of the previous process, one of the following assertions must be true:

(a) ress(S) < 1; or (b) there exists a sequence of positive disjoint elements {a, } in E such that

S(Eal i Ea2 *)Ea,) C Ea, Ea2 Ean,

San >an, lIanIl = 1, an>O VneN, an l am if n $ m.

Suppose that (b) holds. In this case, we take the class a7 of the sequence {an } in the Banach lattice F- = l(E)/s(E). Since 0 < a < Sn&Vn e N and r(S) < 1 (note that S e (D')), & must be equal to 0, whence {an} e s(E) . Therefore, given 0 < e < 1, there exist Yn, Zn, x e E such that

an = Yn + Zn, ? < Yn < x Vn e N, sup II Zn < E. n

As {Yn } is an order bounded disjoint sequence, we get limn IIYn I = 0 (see [1, Theorem 12.13]). So limsupn IIanII <_ < 1, which is a contradiction.

Corollary 2.7. Let E be an ALp-space with 1 < p < oo or an AM-space. Let S be a positive operator on E such that r(S) < 1. If S satisfies the Doeblin condition then re,,(S) < 1.

Proof. Since S' satisfies conditions of Theorem 2.6, it follows that r,,,(S')=

ress(S) < 1.

Note that every quasi-compact operator T satisfies the Doeblin conditions and also that when 0 < S < T, S e (D) (resp. (D')) if T e (D) (resp. (D')). The following corollary is then immediate.

Corollary 2.8. Let E be an ALp-space with 1 < p < oo or an AM-space, and let S, T e 2(E) be positive operators such that 0 < S < T. If ress(T) < r(S) then ress(S) < r(S). Remark 2.9. It is unknown if Corollary 2.8 holds for arbitrary Banach lattices.




3. QUASI-COMPACTNESS OF DOMINATED POSITIVE C0-SEMIGROUPS

In this section we are going to apply the results of the above section to positive Co-semigroups. We recall first some standard definitions and notations (see [6]).

Let {T(t) : t > O} be a Co-semigroup of bounded linear operators on the Ba- nach space E with generator A. The growth bound co(A) of the Co-semigroup {T(t)} is defined by

w(A) = inf{O e R: 3M e R+ such that IIT(t)II < Me" Vt > O}

= lim I

log 11 T(t)tj t t

and the essential bound, co,ess(A), by

(ess (A) = lim I

log II T(t) 1 t t

It is known that

r(T(t)) = et@(A) and ress(T(t)) = etwess(A)

These two bounds are related via

co(A) = max(wOess(A), s(A))

where s(A) is the spectral bound of the generator A, that is,

s(A) = sup{Re:)i e a(A)}.

When dealing with Co-semigroups, Theorem 2.6 can be rewritten as

Theorem 3.1. Let E be an ALp-space with 1 < p < oc. Let {T(t)} be a positive Co-semigroup on E with generator A. Assume that there exist to ? 0, u eE+, and i < 1 such that

T(tO)BE C [-U , U] + rlBE-

Then, if c(A) < 0, {T(t)} is quasi-compact, i.e., wOess(A) < 0.

Analogously, it can also be proved:

Theorem 3.2. Let E be an ALp-space with 1 < p < oo or an AM-space. Let {T(t)} be a positive Co-semigroup on E with generator A. Assume that there exist to> 0, 0 <,u e E', and il < 1 such that

II T(to)f /1 < A(f) + illfjjj Vf E E+.

Then, if w(A) < 0, T(t) is quasi-compact, i.e., w)ess(A) < 0.

ACKNOWLEDGMENT

The author wishes to thank Professors F. Andreu, J. Maz6n, and F. Rabiger for many stimulating conversations concerning the subject of this paper.



426 JOSEP MARTINEZ

REFERENCES

1. C. D. Aliprantis and 0. Burkinshaw, Positive operators, Academic Press, New York, 1985.

2. F. Andreu, V. Caselles, J. Martinez, and J. M. Maz6n, The essential spectrum ofAM-compact operators, Indag. Math. (N.S.) 2 (1991), 149-158.

3. V. Caselles, On the peripherical spectrum of positive operators, Israel J. Math. 58 (1987), 144-160.

4. P. G. Dodds and D. H . Fremlin, Compact operators in Banach lattices, Israel J. Math. 34

(1979), 287-320.

5. H. P. Lotz, Positive linear operators on LP and Doeblin condition, Aspects of Positivity in Functional Analysis (R. Nagel, U. Schottebeck, and M. Wolff, eds.), North-Holland, Amsterdam, 1986, pp. 137-156.

6. J. Martinez and J. M. Maz6n, Quasicompactness of dominated positive operators and Co- semigroups, Math. Z. 207 (1991), 109-120.

7. R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin and New York, 1986.

8. B. de Pagter and A. R. Schep, Measures of non-compactness of positive operators in Banach lattices, J. Funct. Anal. 78 (1988), 31-35.

9. H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974.

10. L. Weis and M. Wolff, On the essential spectrum of operators on L1, Semesterbericht Funk. Anal., Tiubingen, Sommersemester, 1984, 103-112.

11. A. C. Zaanen, Riesz spaces. II, North-Holland, Amsterdam, 1983.

DEPARTMENTO DE ANALISIS MATEMATICO, UNIVERSITAT DE VALENCIA, DR. MOLINER, 50, 46100 BURJASSOT, SPAIN



the essential spectral radius of dominated positive operators

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