Research Article Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

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1 Abstract and Applied Analysis Volume 2011, Article ID , 13 pages doi: /2011/ Research Article Essential Norm of Composition Operators on Banach Spaces of Hölder Functions A. Jiménez-Vargas, 1 Miguel Lacruz, 2 and Moisés Villegas-Vallecillos 1 1 Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Almería, Spain 2 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina, Mercedes s/n, Sevilla, Spain Correspondence should be addressed to Miguel Lacruz, lacruz@us.es Received 29 June 2011; Revised 30 September 2011; Accepted 30 September 2011 Academic Editor: Simeon Reich Copyright q 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. Let X, d be a pointed compact metric space, let 0 < α < 1, and let ϕ : X X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator C ϕ induced by the symbol ϕ on the spaces lip 0 and Lip 0 is given by the formula C ϕ e d ϕ x,ϕ y α /d x, y α whenever the dual space lip 0 has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space. 1. Introduction Let X, d be a compact metric space with a distinguished point e X and 0 <α<1. The formula d α x, y d x, y α defines a new metric on X, and the metric space is said to be a Hölder metric space of order α. As usual, K denotes the field of real or complex numbers. The Lipschitz space Lip 0 is the Banach space of all Lipschitz functions f : X K on the Hölder metric space for which f e 0 under the standard Lipschitz norm L α f ) { f x f y ) } : x, y X, x / y. 1.1 Notice that the Lipschitz functions on are precisely the Hölder functions of order α on X, d.

2 2 Abstract and Applied Analysis The little Lipschitz space lip 0 is the closed subspace consisting of those functions f Lip 0 that satisfy the following local flatness condition: f x f y ) The Lipschitz space Lip 0 has a canonical predual F,thefree 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 δ x f ) f x, x X, f Lip0 ), 1.3 in the dual space Lip 0. As it turns out, F is itself the dual space of lip 0. 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 L E the algebra of all bounded linear operators on a Banach space E, and by K E, the closed ideal of all compact operators on E. Theessential norm T e of an operator T L E is ust the distance from T to K E, thatis, T e inf{ T K : K K E }. 1.4 It is clear that an operator T L E is compact if and only if T e 0. Recall that a Banach space E is said to have the approximation property if the identity operator on E can be approximated uniformly on every compact subset of E by operators of finite rank. Let X, d be a pointed compact metric space with base point e X, let0<α<1, and let ϕ : X X be a Lipschitz mapping that preserves the base point, that is, ϕ e e and L ϕ ) { )) } d ϕ x, ϕ y : d x, y ) : x, y X, x / y <. 1.5 The composition operator C ϕ : lip 0 lip 0 is defined by the expression Cϕ f ) x f ϕ x ), x X, f lip0 X,d α ). 1.6 The aim of this paper is to give lower and upper estimates for the essential norm of the composition operator C ϕ on lip 0 in terms of ϕ. Results along these lines were obtained by Montes-Rodríguez 3, 4 and more recently by Galindo and Lindström 5, andalsoby Galindo et al. 6.InSection 2, we compute the norm of C ϕ. We show that d ϕ x, ϕ y )) α Cϕ x / y. 1.7

3 Abstract and Applied Analysis 3 The prototype of a formula as above with α 1 was provided by Weaver in 1, Proposition for the composition operator C ϕ on the space Lip 0 X, d. InSection 3, we give a lower bound for the essential norm of the operator C ϕ, namely, d ϕ x, ϕ y )) α C ϕ e. 1.8 Section 4 contains our main result. When the dual space lip 0 has the approximation property, we show that C ϕ e d ϕ x,ϕ y )) α. 1.9 The proof of this inequality depends on some results involving shrinking compact approximating sequences on lip 0. Using the fact that the space Lip 0 is isometrically isomorphic to the second dual of lip 0, and the relationship between the essential norm of an operator and its adoint, we derive in Section 5 the same formula for the essential norm of the operator C ϕ on Lip 0. It is natural to ask for some examples where the dual space lip 0 has the approximation property. For instance, this happens if X is uniformly discrete, that is, inf x / y d x, y > 0 see 2, Proposition 4.4. Also, lip 0 has the approximation property whenever the space lip 0 is isomorphic to c 0 and hence Lip 0 is isomorphic to l and F is isomorphic to l 1. A classical result of Bonic et al. 7 ensures that lip 0 is isomorphic to c 0 whenever X 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 X of a Hilbert space containing the origin has the property that lip 0 is isomorphic to c 0 if and only if X 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 0 < α 1/2 2, Theorem 8.5 and for any Banach space that has nontrivial Rademacher type if 0 < α < 1 2, Theorem 8.4. Kalton conectured that this holds in full generality for all Banach spaces. Let us recall now that a metric space X, d satisfies the doubling condition or has finite Assouad dimension if there is an integer n such that for any δ > 0, every closed ball of radius δ can be covered by at most n closed balls of radius δ/2. A theorem of Assouad 8 asserts that whenever a metric space X, d satisfies the doubling condition, every Hölder metric space Lipschitz embeds in the euclidean space R n. Using this result, Kalton observed that if a compact metric space X, d satisfies the doubling condition, then the space lip 0 is isomorphic to c 0 2, Theorem 6.5. Furthermore, he also showed that the converse is false by means of a counterexample 2, Proposition 6.8.

4 4 Abstract and Applied Analysis 2. The Norm of C ϕ on Lip 0 The aim of this section is to derive a formula for the norm of the composition operator C ϕ on lip 0 in terms of the Lipschitz constant of ϕ. A similar expression was already provided by Weaver for the composition operator C ϕ on the space Lip 0 X, d, obtaining in 1, Proposition the following identity: d ϕ x, ϕ y )) C ϕ x / y d x, y ). 2.1 Theorem 2.1. Let X be a pointed compact metric space, 0 < α < 1 and ϕ : X X abase point preserving Lipschitz mapping. Then the norm of the composition operator C ϕ : lip 0 lip 0 is given by the expression d ϕ x, ϕ y )) α C ϕ x / y. 2.2 Proof. We follow the steps of the proof of Weaver s formula. One inequality is formally identical, while the other inequality needs an adustment of the suitable attaining functions. For any f lip 0 with L α f 1, we have and so L α Cϕ f ) x / y f ϕ x ) f ϕ y )) ϕ x / ϕ y ) )) f ϕ x f ϕ y d ϕ x,ϕ y )) α ) d ϕ x,ϕ y )) α L α f x / y d ϕ x,ϕ y )) α x / y, C ϕ L α f 1 d ϕ x,ϕ y )) α x / y 2.3 L α Cϕ f ) d ϕ x,ϕ y )) α x / y. 2.4 For the converse inequality, fix two points x, y X such that ϕ x / ϕ y and choose β strictly between α and 1. Define h : X R by h z d z, ϕ y )) β ) β d z, ϕ x 2d ϕ x, ϕ y )), 2.5 β α and f : X R by f z h z h e. 2.6

5 Abstract and Applied Analysis 5 It is not hard to show that f lip 0 with L α f 1 see, for instance, 9,so Cϕ Lα Cϕ f ) ) )) f ϕ x f ϕ y d )) α ϕ x, ϕ y. 2.7 Taking remum over x and y, we conclude that d ϕ x, ϕ y )) α Cϕ x / y The Lower Estimate of the Essential Norm of C ϕ on Lip 0 Next we bound from below the essential norm of C ϕ on lip 0 by means of an asymptotic quantity that measures the local flatness of ϕ. Theorem 3.1. Let X be a pointed compact metric space, 0 < α < 1 and ϕ : X X abase point preserving Lipschitz mapping. Then the essential norm of the operator C ϕ : lip 0 lip 0 satisfies the lower estimate d ϕ x, ϕ y )) α C ϕ e. 3.1 We will need the following description of the weak convergence in lip 0.This result is part of the folklore and it is immediate from the Banach-Steinhaus theorem since lip 0 span{δ x : x X} see Weaver 1, Theorem Lemma 3.2. Let X be a pointed compact metric space, 0 <α<1 and {f n } a sequence in lip 0. Then {f n } converges to 0 weakly in lip 0 if and only if {f n } is bounded in lip 0 and converges to 0 pointwise on X. Proof of Theorem 3.1. Since the mapping ϕ : X X is Lipschitz, the function t d ϕ x, ϕ y )) α, t >0 3.2 is well defined. It is easy to check that d ϕ x, ϕ y )) α inf t>0 d ϕ x, ϕ y )) α. 3.3

6 6 Abstract and Applied Analysis Now, for every natural number n we can take a real number t n such that 0 <t n < 2/n 1 L ϕ α 1/α and two points x n,y n X such that 0 <d x n,y n <t n, satisfying )) α d ϕ x,ϕ y n < inf t>0 d ϕ x,ϕ y )) α 1 n, )) α d ϕ x,ϕ y n 1 n < d ϕ xn,ϕ )) α y n d ) α. x n,y n 3.4 In this way we obtain two sequences {x n } and {y n } in X such that 0 <d [ ] 1/α ) 2 x n,y n < n 1 L ϕ ) α), d ϕ x n,ϕ )) α y n d d ϕ x, ϕ y )) α ) α inf x n,y n t>0 < 1 n, 3.5 for all n N, and this last inequality implies that inf t>0 d ϕ x, ϕ y )) α d ϕ xn,ϕ )) α y n n d ) α. 3.6 x n,y n Now take x n,y n X for which ϕ x n / ϕ y n and choose β α, 1. Define h n,f n : X R by h n x d x, ϕ y n )) β d x, ϕ xn ) β 2d ϕ x n,ϕ y n )) β α, 3.7 f n x h n x h n e. Using 3.5, we have h n 1/n. Clearly, f n lip 0 with L α f n L α h n 1and f n 2/n for all n N. Moreover, an easy calculation shows that f n ϕ x n f n ϕ y n d ϕ x n,ϕ y n α. Since d ϕ x n,ϕ y n )) α d x n,y n ) α f n ϕ xn ) f n ϕ yn )) d x n,y n ) α L α Cϕ f n ), 3.8 for all n N, we have d ϕ x n,ϕ )) α y n n d ) α x n,y n ) L α Cϕ f n. n 3.9

7 Abstract and Applied Analysis 7 By Lemma 3.2, {f n } 0 weakly in lip 0.Thus,ifK is any compact operator from lip 0 into lip 0, then n L α Kf n 0 because compact operators map weakly convergent sequences into norm convergent sequences. It follows that ) ) )) L α Cϕ f n Lα Cϕ f n Lα Kfn n n L α Cϕ K ) ) f n n 3.10 C ϕ K. Combining 3.3, 3.6, 3.9, and 3.10, we conclude that d ϕ x,ϕ y )) α C ϕ K By taking the infimum on both sides of this inequality over all compact operators K on lip 0, we obtain the lower estimate d ϕ x,ϕ y )) α C ϕ e The Upper Estimate of the Essential Norm of C ϕ on Lip 0 Now we prove that the lower bound of the essential norm of C ϕ on lip 0 obtained in Section 3 is also an upper bound whenever the dual space lip 0 has the approximation property. Theorem 4.1. Let X be a pointed compact metric space and 0 <α<1. Suppose that the dual space lip 0 has the approximation property. Let ϕ : X X be a base point preserving Lipschitz mapping. Then the essential norm of the composition operator C ϕ : lip 0 lip 0 satisfies C ϕ e d ϕ x, ϕ y )) α. 4.1 The strategy for the proof of Theorem 4.1 is to work with a sequence {K n } of compact operators on lip 0 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 {K n } is called a compact approximating sequence for a separable Banach space E if each K n : E E is a compact operator and n I K n f 0 for every f E, where I denotes the identity operator on E. Also, we say that {K n } is shrinking if n I K n f 0 for every f E. Johnson 10, Theorem 2 showed that both the Banach space lip 0 and its dual space lip 0 are separable.

8 8 Abstract and Applied Analysis On the other hand, the Banach-Mazur distance between isomorphic Banach spaces E, F is defined by { d E, F inf T T 1 : T is an isomorphism of E onto F }. 4.2 We say that E embeds almost isometrically into F provided that for every ε > 0, there is a subspace F ε F such that d E, F ε < 1 ε. The next proposition is immediate from a result of Kalton 11. Proposition 4.2 see 11, Corollary 3. Let {K n } be a sequence of compact operators between Banach spaces E and F and let us pose that n K nf,f 0 for all f F and f E. Then there exists a sequence {K c n} of compact operators such that K c n conv{k m : m n} and n K c n 0. Lemma 4.3. Let X be a pointed compact metric space and 0 < α < 1. Suppose that the dual space lip 0 has the approximation property. Then there is a shrinking compact approximating sequence {K n } on lip 0 such that n I K n 1. Proof. Since the dual space lip 0 is separable and has the approximation property, it has the metric approximation property and therefore there is a shrinking compact approximating sequence {S n } on lip 0. We claim that for every N, there exist a natural n and a compact operator K on lip 0 in the convex hull of the set {S m : m n } such that I K < 1 1/ 2. Fix N. Now, for the proof of our claim, we use a result that ensures that lip 0 embeds almost isometrically into c 0. 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 F c 0 and an isomorphism T : lip 0 F such that T T 1 < 1 1/. Now consider M n : T S n T 1, and notice that {M n } is a shrinking compact approximating sequence on F. Next, let {P n } be the sequence of proections on c 0 defined by P n x k x k, 1 k n, P n x k 0, k >n, 4.3 for all x c 0,andletJ : F c 0 be the inclusion map. Then consider the sequence of compact operators D n : P n J JM n defined from F into c 0. Notice that n Dnf,f 0for all f c 0 and f F. It follows from Proposition 4.2 that there is a sequence of operators {Dn} c such that Dn c conv{d m : m n} and n Dn c 0. This gives rise to a pair of shrinking compact approximating sequences {Pn} c and {Mn} c such that, for each n N, Pn c conv{p m : m n}, Mn c conv{m m : m n}, and n PnJ c JMn c 0. Now, consider L n : T 1 MnT c. We have I L n I T 1 MnT c T 1 T I M c n 1 1 ) I M c n 1 1 ) J I M c n

9 Abstract and Applied Analysis ) I P c n J PnJ c JMn c 1 1 ) 1 P c nj JMn, c 4.4 for all n N. Finally, choose n large enough so that Pn c J JMn c < 1/ and conclude that I L n < 1 1/ 2. The claim is proved if we take K : L n. The proof of the lemma will be finished if we show that {K } is a shrinking compact approximating sequence on lip 0.Letf lip 0 and ε>0. Since n L α f S n f 0, there exists m 0 N such that L α f S n f <εfor n m 0.If m 0,usingthat K conv{s m : m n } and n, we conclude that L α f K f <ε. Hence L α f K f 0. This shows that {K } is approximating on lip 0 and similarly it is seen that {K } is shrinking. There is another preinary result that is needed for the proof of Theorem 4.1 and that can be stated as follows. Lemma 4.4. Let X be a pointed compact metric space, 0 < α < 1 and ϕ : X X abase point preserving Lipschitz mapping. Let {K n } be a shrinking compact approximating sequence on lip 0. Then, for each t>0, n L α f 1 d x, y t [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) Proof. Fix t>0. Since the inequality f diam X α L α f is satisfied for all f lip 0, there is a continuous inection J : lip 0,L α lip 0,. Moreover, it follows from the Arzelà-Ascoli Theorem that J is a compact operator, and by Schauder s theorem, its adoint J is a compact operator, too. Let B be the unit ball of lip 0,. Since the bounded sequence of operators { I K n } converges to zero pointwise on lip 0 and J B is a relatively compact set in lip 0,L α, it follows that n I K n J 0. Thus, n J I K n 0. Let ε>0begiven and choose m N such that if n m, then I K n f <εt α /4 for all f lip 0 with L α f 1. For n m, we have [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) 2 I Kn f t α < ε 2, 4.6 whenever f lip 0 with L α f 1andx, y X such that d x, y t. Finally, we get L α f 1 d x, y t [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) <ε, 4.7 as required.

10 10 Abstract and Applied Analysis We now are ready to prove our main result. Proof of Theorem 4.1. Let {K n } be the sequence of operators on lip 0 provided by Lemma 4.3. Since each K n is a compact operator, so is the product C ϕ K n : lip 0 lip 0 and therefore C ϕ e C ϕ I K n. 4.8 Next, fix t>0 and notice that C ϕ I K n L α f 1 L α f 1 L α Cϕ I K n f ) x / y L α f 1 [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) L α f 1 d x, y t [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) [ I K n f ] ϕ x ) [ I K n f ] ϕ y )). 4.9 Then, for every f lip 0, we have [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) ϕ x / ϕ y L α [ I Kn f ] [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) d ϕ x, ϕ y )) α d ϕ x,ϕ y )) α, d ϕ x,ϕ y )) α 4.10 so that L α f 1 0<d x,y <t [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) I K n 0<d x,y <t d ϕ x,ϕ y )) α. 4.11

11 Abstract and Applied Analysis 11 Now, combining the above inequalities, we obtain C e ϕ I K n L α f 1 d x, y t d ϕ x,ϕ y )) α [ I K n f ] ϕ x ) [ I K n f ] ϕ y )) Letting n and using Lemmas 4.3 and 4.4, we conclude that Cϕ e d ϕ x,ϕ y )) α Finally, taking its as yields the desired inequality. 5. The Essential Norm of C ϕ on Lip 0 Now we extend the estimates on the essential norm of a composition operator to the spaces Lip 0. Recall that lip 0 is isometrically isomorphic to Lip 0 whenever X, d is a pointed compact metric space and α 0, 1 see 1, Theorem and Proposition As a matter of fact, the mapping Δ : lip 0 Lip 0, defined by Δ F x F δ x, F lip0,x X ), 5.1 is an isometric isomorphism. If T is a bounded linear operator on a Banach space, then T T. However, this identity is no longer true for the essential norm. Since the adoint of a compact operator is again a compact operator, we always have T e T e and therefore T e T e T e. Axler et al. 12 showed that in fact T e T e, but they gave a counterexample where T e < T e. Theorem 5.1. Let X be a pointed compact metric space, 0 <α<1 and ϕ : X X a point preserving Lipschitz mapping. Then the essential norm of the operator C ϕ :Lip 0 Lip 0 satisfies the lower estimate Cϕ e d ϕ x,ϕ y )) α. 5.2 If, in addition, the space lip 0 has the approximation property, then one has the upper estimate C ϕ e d ϕ x,ϕ y )) α. 5.3

12 12 Abstract and Applied Analysis Proof. Let us start with the lower estimate. Let {f n } be the weakly null sequence in lip 0 that we constructed for the proof of Theorem 3.1. Then the sequence {f n } is weakly null in Lip 0.Thus,ifK is any compact operator on Lip 0, we have n L α Kf n 0. Hence, the same computation we performed in Theorem 3.1 yields the lower estimate d ϕ x, ϕ y )) α Cϕ e. 5.4 Now, for the upper estimate, given F lip 0 and x X, noticethat Δ 1 C ϕ Δ ) F δ x C ϕ Δ ) F ) x C ϕ Δ F x Δ F ϕ x ) F ) ) δ ϕ x F δ x C ϕ lip0 C ϕ lip0 ) F δx. ) ) F C ϕ lip0 δ x 5.5 Since lip 0 span{δ x : x X}, we conclude that Δ 1 C ϕ Δ C ϕ lip0. Finally, using the relationship between the essential norm of an operator and that of its second adoint, and applying Theorem 4.1,weget C e ϕ Δ 1 C ϕ Δ e e C ϕ lip0 X, d ) C α ϕ lip0 e d ϕ x,ϕ y )) α, 5.6 as we wanted. Acknowledgments The authors would like to thank the referees for valuable criticism and lots of useful input. This paper was partially ported by Junta de Andalucía under Grant FQM The first and third authors were partially ported by MICINN under Proect MTM , and the second author under Proect MTM This paper is dedicated to the memory of Nigel J. Kalton. References 1 N. Weaver, Lipschitz Algebras, World Scientific Publishing, River Edge, NJ, USA, N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collectanea Mathematica, vol. 55, no. 2, pp , A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific Journal of Mathematics, vol. 188, no. 2, pp , A. Montes-Rodríguez, Weighted composition operators on weighted Banach spaces of analytic functions, the London Mathematical Society., vol. 61, no. 3, pp , P. Galindo and M. Lindström, Essential norm of operators on weighted Bergman spaces of infinite order, Operator Theory, vol. 64, no. 2, pp , 2010.

13 Abstract and Applied Analysis 13 6 P. Galindo, M. Lindström, and S. Stević, Essential norm of operators into weighted-type spaces on the unit ball, Abstract and Applied Analysis, vol. 2011, Article ID , R. Bonic, J. Frampton, and A. Tromba, Λ-Manifolds, Functional Analysis, vol. 3, no. 2, pp , P. Assouad, Plongements lipschitziens dans R n, Bulletin de la Société Mathématique de France, vol. 111, no. 4, pp , E. Mayer-Wolf, Isometries between Banach spaces of Lipschitz functions, Israel Mathematics, vol. 38, no. 1-2, pp , J. A. Johnson, Lipschitz spaces, Pacific Mathematics, vol. 51, pp , N. J. Kalton, Spaces of compact operators, Mathematische Annalen, vol. 208, pp , S. Axler, N. Jewell, and A. Shields, The essential norm of an operator and its adoint, Transactions of the American Mathematical Society, vol. 261, no. 1, pp , 1980.

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