`Abstract and Applied AnalysisVolume 2011, Article ID 590853, 13 pageshttp://dx.doi.org/10.1155/2011/590853`
Research Article

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

1Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina, Mercedes s/n, 41012 Sevilla, Spain

Received 29 June 2011; Revised 30 September 2011; Accepted 30 September 2011

Copyright © 2011 A. Jiménez-Vargas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a pointed compact metric space, let , and let be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator induced by the symbol on the spaces and is given by the formula whenever the dual space has the approximation property. This happens in particular when is an infinite compact subset of a finite-dimensional normed linear space.

1. Introduction

Let be a compact metric space with a distinguished point and . The formula defines a new metric on , and the metric space is said to be a Hölder metric space of order . As usual, denotes the field of real or complex numbers.

The Lipschitz space is the Banach space of all Lipschitz functions on the Hölder metric space for which under the standard Lipschitz norm Notice that the Lipschitz functions on are precisely the Hölder functions of order on .

The little Lipschitz space is the closed subspace consisting of those functions that satisfy the following local flatness condition:

The Lipschitz space has a canonical predual , the free Lipschitz space on , also known as the Arens-Eells space in [1], that can be defined as the closed linear span of the point evaluations in the dual space . As it turns out, is itself the dual space of . The structure of spaces of Lipschitz and Hölder functions and their preduals on general metric spaces was studied by Kalton in [2]. We refer to the book [1] by Weaver for a complete study of the spaces of Lipschitz functions.

We denote by the algebra of all bounded linear operators on a Banach space , and by , the closed ideal of all compact operators on . The essential norm of an operator is just the distance from to , that is, It is clear that an operator is compact if and only if .

Recall that a Banach space is said to have the approximation property if the identity operator on can be approximated uniformly on every compact subset of by operators of finite rank.

Let be a pointed compact metric space with base point , let , and let be a Lipschitz mapping that preserves the base point, that is, and The composition operator is defined by the expression

The aim of this paper is to give lower and upper estimates for the essential norm of the composition operator on in terms of . Results along these lines were obtained by Montes-Rodríguez [3, 4] and more recently by Galindo and Lindström [5], and also by Galindo et al. [6]. In Section 2, we compute the norm of . We show that The prototype of a formula as above with was provided by Weaver in [1, Proposition  1.8.2] for the composition operator on the space . In Section 3, we give a lower bound for the essential norm of the operator , namely, Section 4 contains our main result. When the dual space has the approximation property, we show that The proof of this inequality depends on some results involving shrinking compact approximating sequences on . Using the fact that the space is isometrically isomorphic to the second dual of , and the relationship between the essential norm of an operator and its adjoint, we derive in Section 5 the same formula for the essential norm of the operator on .

It is natural to ask for some examples where the dual space has the approximation property. For instance, this happens if is uniformly discrete, that is, (see [2, Proposition  4.4]). Also, has the approximation property whenever the space is isomorphic to and hence is isomorphic to and is isomorphic to .

A classical result of Bonic et al. [7] ensures that is isomorphic to whenever is an infinite compact subset of a finite-dimensional normed linear space. This result was corrected by Weaver, who asked whether such an isomorphism could be extended to any compact metric space [1, page 98]. Kalton answered this question negatively by proving that a compact convex subset of a Hilbert space containing the origin has the property that is isomorphic to if and only if is finite dimensional [2, Theorem  8.3]. In fact this statement is true for every general Banach space in place of a Hilbert space if [2, Theorem  8.5] and for any Banach space that has nontrivial Rademacher type if [2, Theorem  8.4]. Kalton conjectured that this holds in full generality for all Banach spaces.

Let us recall now that a metric space satisfies the doubling condition (or has finite Assouad dimension) if there is an integer such that for any , every closed ball of radius can be covered by at most closed balls of radius . A theorem of Assouad [8] asserts that whenever a metric space satisfies the doubling condition, every Hölder metric space Lipschitz embeds in the euclidean space . Using this result, Kalton observed that if a compact metric space satisfies the doubling condition, then the space is isomorphic to [2, Theorem  6.5]. Furthermore, he also showed that the converse is false by means of a counterexample [2, Proposition  6.8].

2. The Norm of on

The aim of this section is to derive a formula for the norm of the composition operator on in terms of the Lipschitz constant of . A similar expression was already provided by Weaver for the composition operator on the space , obtaining in [1, Proposition  1.8.2] the following identity:

Theorem 2.1. Let be a pointed compact metric space, and a base point preserving Lipschitz mapping. Then the norm of the composition operator is given by the expression

Proof. We follow the steps of the proof of Weaver’s formula. One inequality is formally identical, while the other inequality needs an adjustment of the suitable attaining functions. For any with , we have and so For the converse inequality, fix two points such that and choose strictly between and 1. Define by and by It is not hard to show that with (see, for instance, [9]), so Taking supremum over and , we conclude that

3. The Lower Estimate of the Essential Norm of on

Next we bound from below the essential norm of on by means of an asymptotic quantity that measures the local flatness of .

Theorem 3.1. Let be a pointed compact metric space, and a base point preserving Lipschitz mapping. Then the essential norm of the operator satisfies the lower estimate

We will need the following description of the weak convergence in . This result is part of the folklore and it is immediate from the Banach-Steinhaus theorem since (see Weaver [1, Theorem  3.3.3]).

Lemma 3.2. Let be a pointed compact metric space, and a sequence in . Then converges to 0 weakly in if and only if is bounded in and converges to 0 pointwise on .

Proof of Theorem 3.1. Since the mapping is Lipschitz, the function is well defined. It is easy to check that Now, for every natural number we can take a real number such that and two points such that , satisfying In this way we obtain two sequences and in such that for all , and this last inequality implies that Now take for which and choose . Define by Using (3.5), we have . Clearly, with and for all . Moreover, an easy calculation shows that . Since for all , we have By Lemma 3.2, weakly in . Thus, if is any compact operator from into , then because compact operators map weakly convergent sequences into norm convergent sequences. It follows that Combining (3.3), (3.6), (3.9), and (3.10), we conclude that By taking the infimum on both sides of this inequality over all compact operators on , we obtain the lower estimate

4. The Upper Estimate of the Essential Norm of on

Now we prove that the lower bound of the essential norm of on obtained in Section 3 is also an upper bound whenever the dual space has the approximation property.

Theorem 4.1. Let be a pointed compact metric space and . Suppose that the dual space has the approximation property. Let be a base point preserving Lipschitz mapping. Then the essential norm of the composition operator satisfies

The strategy for the proof of Theorem 4.1 is to work with a sequence of compact operators on that satisfies some prescribed conditions that are stated in Lemma 4.3 below. We borrow this technique from the work of Montes-Rodríguez [3].

First we recall some notions and results. A sequence is called a compact approximating sequence for a separable Banach space if each is a compact operator and for every , where denotes the identity operator on . Also, we say that is shrinking if for every .

Johnson [10, Theorem  2] showed that both the Banach space and its dual space are separable.

On the other hand, the Banach-Mazur distance between isomorphic Banach spaces is defined by We say that embeds almost isometrically into provided that for every , there is a subspace such that .

The next proposition is immediate from a result of Kalton [11].

Proposition 4.2 (see [11, Corollary  3]). Let be a sequence of compact operators between Banach spaces and and let us suppose that for all and . Then there exists a sequence of compact operators such that and .

Lemma 4.3. Let be a pointed compact metric space and . Suppose that the dual space has the approximation property. Then there is a shrinking compact approximating sequence on such that .

Proof. Since the dual space is separable and has the approximation property, it has the metric approximation property and therefore there is a shrinking compact approximating sequence on . We claim that for every , there exist a natural and a compact operator on in the convex hull of the set such that .
Fix . Now, for the proof of our claim, we use a result that ensures that embeds almost isometrically into . We refer to Kalton [2, Theorem  6.6] for a simple proof of this result due to Yoav Benyamini. Thus, there is a closed subspace and an isomorphism such that . Now consider , and notice that is a shrinking compact approximating sequence on . Next, let be the sequence of projections on defined by for all , and let be the inclusion map. Then consider the sequence of compact operators defined from into . Notice that for all and . It follows from Proposition 4.2 that there is a sequence of operators such that and . This gives rise to a pair of shrinking compact approximating sequences and such that, for each ,, , and . Now, consider . We have for all . Finally, choose large enough so that and conclude that . The claim is proved if we take .
The proof of the lemma will be finished if we show that is a shrinking compact approximating sequence on . Let and . Since , there exists such that for . If , using that and , we conclude that . Hence . This shows that is approximating on and similarly it is seen that is shrinking.

There is another preliminary result that is needed for the proof of Theorem 4.1 and that can be stated as follows.

Lemma 4.4. Let be a pointed compact metric space, and a base point preserving Lipschitz mapping. Let be a shrinking compact approximating sequence on .
Then, for each ,

Proof. Fix . Since the inequality is satisfied for all , there is a continuous injection . Moreover, it follows from the Arzelà-Ascoli Theorem that is a compact operator, and by Schauder's theorem, its adjoint is a compact operator, too.
Let be the unit ball of . Since the bounded sequence of operators converges to zero pointwise on and is a relatively compact set in , it follows that . Thus, . Let be given and choose such that if , then for all with . For , we have whenever with and such that . Finally, we get as required.

We now are ready to prove our main result.

Proof of Theorem 4.1. Let be the sequence of operators on provided by Lemma 4.3. Since each is a compact operator, so is the product and therefore Next, fix and notice that Then, for every , we have so that Now, combining the above inequalities, we obtain Letting and using Lemmas 4.3 and 4.4, we conclude that Finally, taking limits as yields the desired inequality.

5. The Essential Norm of on

Now we extend the estimates on the essential norm of a composition operator to the spaces . Recall that is isometrically isomorphic to whenever is a pointed compact metric space and (see [1, Theorem  3.3.3 and Proposition  3.2.2]). As a matter of fact, the mapping , defined by is an isometric isomorphism.

If is a bounded linear operator on a Banach space, then . However, this identity is no longer true for the essential norm. Since the adjoint of a compact operator is again a compact operator, we always have and therefore . Axler et al. [12] showed that in fact , but they gave a counterexample where .

Theorem 5.1. Let be a pointed compact metric space, and a point preserving Lipschitz mapping. Then the essential norm of the operator satisfies the lower estimate If, in addition, the space has the approximation property, then one has the upper estimate

Proof. Let us start with the lower estimate. Let be the weakly null sequence in that we constructed for the proof of Theorem 3.1. Then the sequence is weakly null in . Thus, if is any compact operator on , we have . Hence, the same computation we performed in Theorem 3.1 yields the lower estimate Now, for the upper estimate, given and , notice that Since , we conclude that . Finally, using the relationship between the essential norm of an operator and that of its second adjoint, and applying Theorem 4.1, we get as we wanted.

Acknowledgments

The authors would like to thank the referees for valuable criticism and lots of useful input. This paper was partially supported by Junta de Andalucía under Grant FQM-3737. The first and third authors were partially supported by MICINN under Project MTM 2010-17687, and the second author under Project MTM 2009-08934. This paper is dedicated to the memory of Nigel J. Kalton.

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