Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Abstract

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 lip₀(𝑋,𝑑^𝛼) and Lip₀(𝑋,𝑑^𝛼) is given by the formula ‖𝐶_𝜑‖_𝑒=lim_𝑡→0sup_{0<𝑑(𝑥,𝑦)<𝑡}(𝑑(𝜑(𝑥),𝜑(𝑦))^𝛼/𝑑(𝑥,𝑦)^𝛼) whenever the dual space lip₀(𝑋,𝑑^𝛼)^∗ 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 (𝑋,𝑑^𝛼) 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 Lip₀(𝑋,𝑑^𝛼) is the Banach space of all Lipschitz functions 𝑓∶𝑋→𝕂 on the Hölder metric space (𝑋,𝑑^𝛼) for which 𝑓(𝑒)=0 under the standard Lipschitz norm 𝐿_𝛼𝑓𝑓(𝑥)𝑦=sup−𝑓𝑑𝑥,𝑦^𝛼∶𝑥,𝑦∈𝑋,𝑥≠𝑦. (1.1) Notice that the Lipschitz functions on (𝑋,𝑑^𝛼) are precisely the Hölder functions of order 𝛼 on (𝑋,𝑑).

The little Lipschitz space lip₀(𝑋,𝑑^𝛼) is the closed subspace consisting of those functions 𝑓∈Lip₀(𝑋,𝑑^𝛼) that satisfy the following local flatness condition: lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑓(𝑥)−𝑓𝑦𝑑𝑥,𝑦^𝛼=0. (1.2)

The Lipschitz space Lip₀(𝑋,𝑑^𝛼) has a canonical predual ℱ(𝑋,𝑑^𝛼), the free Lipschitz space on (𝑋,𝑑^𝛼), also known as the Arens-Eells space in [1], that can be defined as the closed linear span of the point evaluations𝛿_𝑥𝑓=𝑓(𝑥),𝑥∈𝑋,𝑓∈Lip₀𝑋,𝑑^𝛼, (1.3) in the dual space Lip₀(𝑋,𝑑^𝛼)^∗. As it turns out, ℱ(𝑋,𝑑^𝛼) is itself the dual space of lip₀(𝑋,𝑑^𝛼). 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. The essential norm ‖𝑇‖_𝑒 of an operator 𝑇∈ℒ(𝐸) is just the distance from 𝑇 to 𝒦(𝐸), that is, ‖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, let 0<𝛼<1, and let 𝜑∶𝑋→𝑋 be a Lipschitz mapping that preserves the base point, that is, 𝜑(𝑒)=𝑒 and 𝐿𝜑𝑑𝜑(𝑥)𝑦∶=sup,𝜑𝑑𝑥,𝑦∶𝑥,𝑦∈𝑋,𝑥≠𝑦<∞. (1.5) The composition operator 𝐶_𝜑∶lip₀(𝑋,𝑑^𝛼)→lip₀(𝑋,𝑑^𝛼) is defined by the expression 𝐶_𝜑𝑓(𝑥)=𝑓𝜑(𝑥),𝑥∈𝑋,𝑓∈lip₀X,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 lip₀(𝑋,𝑑^𝛼) in terms of 𝜑. 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 𝐶_𝜑. We show that 𝐶_𝜑=sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (1.7) The prototype of a formula as above with 𝛼=1 was provided by Weaver in [1, Proposition  1.8.2] for the composition operator 𝐶_𝜑 on the space Lip₀(𝑋,𝑑). In Section 3, we give a lower bound for the essential norm of the operator 𝐶_𝜑, namely, lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐶_𝜑𝑒. (1.8) Section 4 contains our main result. When the dual space lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property, we show that 𝐶_𝜑𝑒≤lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (1.9) The proof of this inequality depends on some results involving shrinking compact approximating sequences on lip₀(𝑋,𝑑^𝛼). Using the fact that the space Lip₀(𝑋,𝑑^𝛼) is isometrically isomorphic to the second dual of lip₀(𝑋,𝑑^𝛼), 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 𝐶_𝜑 on Lip₀(𝑋,𝑑^𝛼).

It is natural to ask for some examples where the dual space lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property. For instance, this happens if 𝑋 is uniformly discrete, that is, inf_𝑥≠𝑦𝑑(𝑥,𝑦)>0 (see [2, Proposition  4.4]). Also, lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property whenever the space lip₀(𝑋,𝑑^𝛼) is isomorphic to 𝑐₀ and hence Lip₀(𝑋,𝑑^𝛼) is isomorphic to ℓ_∞ and ℱ(𝑋,𝑑^𝛼) is isomorphic to ℓ₁.

A classical result of Bonic et al. [7] ensures that lip₀(𝑋,𝑑^𝛼) is isomorphic to 𝑐₀ 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 lip₀(𝑋,𝑑^𝛼) is isomorphic to 𝑐₀ 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 conjectured 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 (𝑋,𝑑^𝛼) 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 lip₀(𝑋,𝑑^𝛼) is isomorphic to 𝑐₀ [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 𝐶_𝜑 on Lip₀(𝑋,𝑑^𝛼)

The aim of this section is to derive a formula for the norm of the composition operator 𝐶_𝜑 on lip₀(𝑋,𝑑^𝛼) 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 Lip₀(𝑋,𝑑), obtaining in [1, Proposition  1.8.2] the following identity: 𝐶_𝜑=sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦𝑑.𝑥,𝑦 (2.1)

Theorem 2.1. Let 𝑋 be a pointed compact metric space, 0<𝛼<1 and 𝜑∶𝑋→𝑋 a base point preserving Lipschitz mapping. Then the norm of the composition operator 𝐶_𝜑∶lip₀(𝑋,𝑑^𝛼)→lip₀(𝑋,𝑑^𝛼) is given by the expression 𝐶_𝜑=sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (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 adjustment of the suitable attaining functions. For any 𝑓∈lip₀(𝑋,𝑑^𝛼) with 𝐿_𝛼(𝑓)≤1, we have 𝐿_𝛼𝐶_𝜑𝑓=sup_𝑥≠𝑦𝑓𝜑(𝑥)−𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼≤sup_{𝜑(𝑥)𝑦≠𝜑}𝑓𝜑(𝑥)−𝑓𝜑𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼⋅sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐿_𝛼𝑓⋅sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼, (2.3) and so 𝐶_𝜑=sup_{𝐿𝛼𝑓≤1}𝐿_𝛼𝐶_𝜑𝑓≤sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (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)=𝑑𝑧,𝜑𝑦^𝛽−𝑑𝑧,𝜑(𝑥)^𝛽2𝑑𝜑(𝑥),𝜑𝑦^𝛽−𝛼, (2.5) and 𝑓∶𝑋→ℝ by 𝑓(𝑧)=ℎ(𝑧)−ℎ(𝑒). (2.6) It is not hard to show that 𝑓∈lip₀(𝑋,𝑑^𝛼) with 𝐿_𝛼(𝑓)=1 (see, for instance, [9]), so 𝐶_𝜑≥𝐿_𝛼𝐶_𝜑𝑓≥𝑓𝜑(𝑥)−𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼=𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (2.7) Taking supremum over 𝑥 and 𝑦, we conclude that 𝐶_𝜑≥sup_𝑥≠𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (2.8)

3. The Lower Estimate of the Essential Norm of 𝐶_𝜑 on Lip₀(𝑋,𝑑^𝛼)

Next we bound from below the essential norm of 𝐶_𝜑 on lip₀(𝑋,𝑑^𝛼) 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 a base point preserving Lipschitz mapping. Then the essential norm of the operator 𝐶_𝜑∶lip₀(𝑋,𝑑^𝛼)→lip₀(𝑋,𝑑^𝛼) satisfies the lower estimate lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐶_𝜑𝑒. (3.1)

We will need the following description of the weak convergence in lip₀(𝑋,𝑑^𝛼). This result is part of the folklore and it is immediate from the Banach-Steinhaus theorem since lip₀(𝑋,𝑑^𝛼)^∗=span{𝛿_𝑥∶𝑥∈𝑋} (see Weaver [1, Theorem  3.3.3]).

Lemma 3.2. Let 𝑋 be a pointed compact metric space, 0<𝛼<1 and {𝑓_𝑛} a sequence in lip₀(𝑋,𝑑^𝛼). Then {𝑓_𝑛} converges to 0 weakly in lip₀(𝑋,𝑑^𝛼) if and only if {𝑓_𝑛} is bounded in lip₀(𝑋,𝑑^𝛼) and converges to 0 pointwise on 𝑋.

Proof of Theorem 3.1. Since the mapping 𝜑∶𝑋→𝑋 is Lipschitz, the function 𝑡⟼sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼,(𝑡>0) (3.2) is well defined. It is easy to check that lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼=inf_𝑡>0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (3.3) 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 sup_{0<𝑑𝑥,𝑦<𝑡𝑛}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼<inf_𝑡>0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼+1𝑛,sup_{0<𝑑𝑥,𝑦<𝑡𝑛}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼−1𝑛<𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛼𝑑𝑥_𝑛,𝑦_𝑛^𝛼. (3.4) In this way we obtain two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 0<𝑑𝑥_𝑛,𝑦_𝑛<2𝑛1+𝐿𝜑^𝛼1/𝛼,𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛼𝑑𝑥_𝑛,𝑦_𝑛^𝛼−inf_𝑡>0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼<1𝑛, (3.5) for all 𝑛∈ℕ, and this last inequality implies that inf_𝑡>0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼=lim_𝑛→∞𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛼𝑑𝑥_𝑛,𝑦_𝑛^𝛼. (3.6) 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)=𝑑𝑥,𝜑𝑦_𝑛^𝛽−𝑑𝑥,𝜑𝑥_𝑛^𝛽2𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛽−𝛼,𝑓_𝑛(𝑥)=ℎ_𝑛(𝑥)−ℎ_𝑛(𝑒). (3.7) Using (3.5), we have ‖ℎ_𝑛‖_∞≤1/𝑛. Clearly, 𝑓_𝑛∈lip₀(𝑋,𝑑^𝛼) with 𝐿_𝛼(𝑓_𝑛)=𝐿_𝛼(ℎ_𝑛)=1 and ‖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 𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛼𝑑𝑥_𝑛,𝑦_𝑛^𝛼=𝑓_𝑛𝜑𝑥_𝑛−𝑓_𝑛𝜑𝑦_𝑛𝑑𝑥_𝑛,𝑦_𝑛^𝛼≤𝐿_𝛼𝐶_𝜑𝑓_𝑛, (3.8) for all 𝑛∈ℕ, we have lim_𝑛→∞𝑑𝜑𝑥_𝑛,𝜑𝑦_𝑛^𝛼𝑑𝑥_𝑛,𝑦_𝑛^𝛼≤limsup_𝑛→∞𝐿_𝛼𝐶_𝜑𝑓_𝑛. (3.9) By Lemma 3.2, {𝑓_𝑛}→0 weakly in lip₀(𝑋,𝑑^𝛼). Thus, if 𝐾 is any compact operator from lip₀(𝑋,𝑑^𝛼)into lip₀(𝑋,𝑑^𝛼), then lim_𝑛→∞𝐿_𝛼(𝐾𝑓_𝑛)=0 because compact operators map weakly convergent sequences into norm convergent sequences. It follows that limsup_𝑛→∞𝐿_𝛼𝐶_𝜑𝑓_𝑛=limsup_𝑛→∞𝐿_𝛼𝐶_𝜑𝑓_𝑛−𝐿_𝛼𝐾𝑓_𝑛≤limsup_𝑛→∞𝐿_𝛼𝐶_𝜑−𝐾𝑓_𝑛≤𝐶_𝜑−𝐾. (3.10) Combining (3.3), (3.6), (3.9), and (3.10), we conclude that lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐶_𝜑−𝐾. (3.11) By taking the infimum on both sides of this inequality over all compact operators 𝐾 on lip₀(𝑋,𝑑^𝛼), we obtain the lower estimate lim_𝑡→0sup_{0<𝑑(𝑥,𝑦)<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐶_𝜑𝑒. (3.12)

4. The Upper Estimate of the Essential Norm of 𝐶_𝜑 on Lip₀(𝑋,𝑑^𝛼)

Now we prove that the lower bound of the essential norm of 𝐶_𝜑 on lip₀(𝑋,𝑑^𝛼) obtained in Section 3 is also an upper bound whenever the dual space lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property.

Theorem 4.1. Let 𝑋 be a pointed compact metric space and 0<𝛼<1. Suppose that the dual space lip₀(𝑋,𝑑^𝛼)^∗ 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∶lip₀(𝑋,𝑑^𝛼)→lip₀(𝑋,𝑑^𝛼) satisfies 𝐶_𝜑𝑒≤lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,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 lip₀(𝑋,𝑑^𝛼) 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 lim_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 lim_𝑛→∞‖(𝐼−𝐾_𝑛)^∗𝑓^∗‖=0 for every 𝑓^∗∈𝐸^∗.

Johnson [10, Theorem  2] showed that both the Banach space lip₀(𝑋,𝑑^𝛼) and its dual space lip₀(𝑋,𝑑^𝛼)^∗ are separable.

On the other hand, the Banach-Mazur distance between isomorphic Banach spaces 𝐸,𝐹 is defined by 𝑑(𝐸,𝐹)=inf‖𝑇‖⋅𝑇⁻¹∶𝑇isanisomorphismof𝐸onto𝐹. (4.2) We say that 𝐸embeds 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 suppose that lim_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 lim_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 lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property. Then there is a shrinking compact approximating sequence {𝐾_𝑛} on lip₀(𝑋,𝑑^𝛼) such that limsup_𝑛→∞‖𝐼−𝐾_𝑛‖≤1.

Proof. Since the dual space lip₀(𝑋,𝑑^𝛼)^∗ 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 lip₀(𝑋,𝑑^𝛼). We claim that for every 𝑗∈ℕ, there exist a natural 𝑛_𝑗≥𝑗 and a compact operator 𝐾_𝑗 on lip₀(𝑋,𝑑^𝛼) in the convex hull of the set {𝑆_𝑚∶𝑚≥𝑛_𝑗} such that ‖𝐼−K_𝑗‖<(1+1/𝑗)².
Fix 𝑗∈ℕ. Now, for the proof of our claim, we use a result that ensures that lip₀(𝑋,𝑑^𝛼) embeds almost isometrically into 𝑐₀. 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_𝑗⊂𝑐₀ and an isomorphism 𝑇_𝑗∶lip₀(𝑋,𝑑^𝛼)→𝐹_𝑗 such that ‖𝑇_𝑗‖⋅‖𝑇_𝑗⁻¹‖<1+(1/𝑗). Now consider 𝑀_𝑛∶=𝑇_𝑗𝑆_𝑛𝑇_𝑗⁻¹, and notice that {𝑀_𝑛} is a shrinking compact approximating sequence on 𝐹_𝑗. Next, let {𝑃_𝑛} be the sequence of projections on 𝑐₀ defined by 𝑃_𝑛𝑥(𝑘)=𝑥(𝑘),(1≤𝑘≤𝑛),𝑃_𝑛𝑥(𝑘)=0,(𝑘>𝑛), (4.3) for all 𝑥∈𝑐₀, and let 𝐽∶𝐹_𝑗↪𝑐₀ be the inclusion map. Then consider the sequence of compact operators 𝐷_𝑛∶=𝑃_𝑛𝐽−𝐽𝑀_𝑛 defined from 𝐹_𝑗 into 𝑐₀. Notice that lim_𝑛→∞⟨𝐷^∗_𝑛𝑓^∗,𝑓^∗∗⟩=0 for all 𝑓^∗∈𝑐^∗₀ and 𝑓^∗∗∈𝐹_𝑗^∗∗. It follows from Proposition 4.2 that there is a sequence of operators {𝐷^𝑐_𝑛} such that 𝐷^𝑐_𝑛∈conv{𝐷_𝑚∶𝑚≥𝑛} and lim_𝑛→∞‖𝐷^𝑐_𝑛‖=0. This gives rise to a pair of shrinking compact approximating sequences {𝑃^𝑐_𝑛} and {𝑀^𝑐_𝑛} such that, for each 𝑛∈ℕ,𝑃^𝑐_𝑛∈conv{𝑃_𝑚∶𝑚≥𝑛}, 𝑀^𝑐_𝑛∈conv{𝑀_𝑚∶𝑚≥𝑛}, and lim_𝑛→∞‖𝑃^𝑐_𝑛𝐽−𝐽𝑀^𝑐_𝑛‖=0. Now, consider 𝐿_𝑛∶=𝑇_𝑗⁻¹𝑀^𝑐_𝑛𝑇_𝑗. We have 𝐼−𝐿_𝑛=𝐼−𝑇_𝑗⁻¹𝑀^𝑐_𝑛𝑇_𝑗≤𝑇_𝑗⁻¹⋅𝑇_𝑗⋅𝐼−𝑀^𝑐_𝑛≤11+𝑗⋅𝐼−𝑀^𝑐_𝑛=11+𝑗⋅𝐽𝐼−𝑀^𝑐_𝑛≤11+𝑗⋅𝐼−𝑃^𝑐_𝑛𝐽+𝑃^𝑐_𝑛𝐽−𝐽𝑀^𝑐_𝑛≤11+𝑗⋅1+𝑃^𝑐_𝑛𝐽−𝐽𝑀^𝑐_𝑛, (4.4) for all 𝑛∈ℕ. Finally, choose 𝑛_𝑗≥𝑗 large enough so that ‖𝑃^𝑐_𝑛𝑗𝐽−𝐽𝑀^𝑐_𝑛𝑗‖<1/𝑗 and conclude that ‖𝐼−𝐿_𝑛𝑗‖<(1+(1/𝑗))². The claim is proved if we take 𝐾_𝑗∶=𝐿_𝑛𝑗.
The proof of the lemma will be finished if we show that {𝐾_𝑗} is a shrinking compact approximating sequence on lip₀(𝑋,𝑑^𝛼). Let 𝑓∈lip₀(𝑋,𝑑^𝛼) and 𝜀>0. Since lim_𝑛→∞𝐿_𝛼(𝑓−𝑆_𝑛𝑓)=0, there exists 𝑚₀∈ℕ such that 𝐿_𝛼(𝑓−𝑆_𝑛𝑓)<𝜀 for 𝑛≥𝑚₀. If 𝑗≥𝑚₀, using that 𝐾_𝑗∈conv{𝑆_𝑚∶𝑚≥𝑛_𝑗} and 𝑛_𝑗≥𝑗, we conclude that 𝐿_𝛼(𝑓−𝐾_𝑗𝑓)<𝜀. Hence lim_𝑗→∞𝐿_𝛼(𝑓−𝐾_𝑗𝑓)=0. This shows that {𝐾_𝑗} is approximating on lip₀(𝑋,𝑑^𝛼) and similarly it is seen that {𝐾_𝑗} 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 𝑋 be a pointed compact metric space, 0<𝛼<1 and 𝜑∶𝑋→𝑋 a base point preserving Lipschitz mapping. Let {𝐾_𝑛} be a shrinking compact approximating sequence on lip₀(𝑋,𝑑^𝛼).
Then, for each 𝑡>0, lim_𝑛→∞sup_{𝐿𝛼𝑓≤1}sup_{𝑑𝑥,𝑦≥𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼=0. (4.5)

Proof. Fix 𝑡>0. Since the inequality ‖𝑓‖_∞≤diam(𝑋)^𝛼⋅𝐿_𝛼(𝑓) is satisfied for all 𝑓∈lip₀(𝑋,𝑑^𝛼), there is a continuous injection 𝐽∶(lip₀(𝑋,𝑑^𝛼),𝐿_𝛼(⋅))↪(lip₀(𝑋,𝑑^𝛼),‖⋅‖_∞). Moreover, it follows from the Arzelà-Ascoli Theorem that 𝐽 is a compact operator, and by Schauder's theorem, its adjoint 𝐽^∗ is a compact operator, too.
Let 𝐵 be the unit ball of (lip₀(𝑋,𝑑^𝛼),‖⋅‖_∞)^∗. Since the bounded sequence of operators {(𝐼−𝐾_𝑛)^∗} converges to zero pointwise on lip₀(𝑋,d^𝛼)^∗ and 𝐽^∗(𝐵) is a relatively compact set in (lip₀(𝑋,𝑑^𝛼),𝐿_𝛼(⋅))^∗, it follows that lim_𝑛→∞‖(𝐼−𝐾_𝑛)^∗𝐽^∗‖=0. Thus, lim_𝑛→∞‖𝐽(𝐼−𝐾_𝑛)‖=0. Let 𝜀>0 be given and choose 𝑚∈ℕ such that if 𝑛≥𝑚, then ‖(𝐼−𝐾_𝑛)𝑓‖_∞<𝜀𝑡^𝛼/4 for all 𝑓∈lip₀(𝑋,𝑑^𝛼) with 𝐿_𝛼(𝑓)≤1. For 𝑛≥𝑚, we have 𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼≤2𝐼−𝐾_𝑛𝑓_∞𝑡^𝛼<𝜀2, (4.6) whenever 𝑓∈lip₀(𝑋,𝑑^𝛼) with 𝐿_𝛼(𝑓)≤1 and 𝑥,𝑦∈𝑋 such that 𝑑(𝑥,𝑦)≥𝑡. Finally, we get sup_{𝐿𝛼𝑓≤1}sup_{𝑑𝑥,𝑦≥𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼<𝜀, (4.7) as required.

We now are ready to prove our main result.

Proof of Theorem 4.1. Let {𝐾_𝑛} be the sequence of operators on lip₀(𝑋,𝑑^𝛼) provided by Lemma 4.3. Since each 𝐾_𝑛 is a compact operator, so is the product 𝐶_𝜑𝐾_𝑛∶lip₀(𝑋,𝑑^𝛼)→lip₀(𝑋,𝑑^𝛼) and therefore 𝐶_𝜑𝑒≤𝐶_𝜑𝐼−𝐾_𝑛. (4.8) Next, fix 𝑡>0 and notice that 𝐶_𝜑𝐼−𝐾_𝑛=sup_{𝐿𝛼𝑓≤1}𝐿_𝛼𝐶_𝜑𝐼−𝐾_𝑛𝑓=sup_{𝐿𝛼𝑓≤1}sup_𝑥≠𝑦𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼≤sup_{𝐿𝛼𝑓≤1}sup_{0<𝑑𝑥,𝑦<𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼+sup_{𝐿𝛼𝑓≤1}sup_{𝑑𝑥,𝑦≥𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼. (4.9) Then, for every 𝑓∈lip₀(𝑋,𝑑^𝛼), we have sup_{0<𝑑𝑥,𝑦<𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼≤sup_{𝜑(𝑥)𝑦≠𝜑}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝜑(𝑥),𝜑𝑦^𝛼⋅sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐿_𝛼𝐼−𝐾_𝑛𝑓⋅sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼, (4.10) so that sup_{𝐿𝛼𝑓≤1}sup_{0<𝑑𝑥,𝑦<𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼≤𝐼−𝐾_𝑛sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (4.11) Now, combining the above inequalities, we obtain 𝐶_𝜑𝑒≤𝐼−𝐾_𝑛sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼+sup_{𝐿𝛼𝑓≤1}sup_{𝑑𝑥,𝑦≥𝑡}𝐼−𝐾_𝑛𝑓𝜑(𝑥)−𝐼−𝐾_𝑛𝑓𝜑𝑦𝑑𝑥,𝑦^𝛼. (4.12) Letting 𝑛→∞ and using Lemmas 4.3 and 4.4, we conclude that 𝐶_𝜑𝑒≤sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (4.13) Finally, taking limits as 𝑡→0 yields the desired inequality.

5. The Essential Norm of 𝐶_𝜑 on Lip₀(𝑋,𝑑^𝛼)

Now we extend the estimates on the essential norm of a composition operator to the spaces Lip₀(𝑋,𝑑^𝛼). Recall that lip₀(𝑋,𝑑^𝛼)^∗∗ is isometrically isomorphic to Lip₀(𝑋,𝑑^𝛼) whenever (𝑋,𝑑) is a pointed compact metric space and 𝛼∈(0,1) (see [1, Theorem  3.3.3 and Proposition  3.2.2]). As a matter of fact, the mapping Δ∶lip₀(𝑋,𝑑^𝛼)^∗∗→Lip₀(𝑋,𝑑^𝛼), defined by Δ(𝐹)(𝑥)=𝐹𝛿_𝑥,𝐹∈lip₀𝑋,𝑑^𝛼∗∗,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 adjoint 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∶Lip₀(𝑋,𝑑^𝛼)→Lip₀(𝑋,𝑑^𝛼) satisfies the lower estimate 𝐶_𝜑𝑒≥lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (5.2) If, in addition, the space lip₀(𝑋,𝑑^𝛼)^∗ has the approximation property, then one has the upper estimate 𝐶_𝜑𝑒≤lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼. (5.3)

Proof. Let us start with the lower estimate. Let {𝑓_𝑛} be the weakly null sequence in lip₀(𝑋,𝑑^𝛼) that we constructed for the proof of Theorem 3.1. Then the sequence {𝑓_𝑛} is weakly null in Lip₀(𝑋,𝑑^𝛼). Thus, if 𝐾 is any compact operator on Lip₀(𝑋,𝑑^𝛼), we have lim_𝑛→∞𝐿_𝛼(𝐾𝑓_𝑛)=0. Hence, the same computation we performed in Theorem 3.1 yields the lower estimate lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,𝑦^𝛼≤𝐶_𝜑𝑒. (5.4) Now, for the upper estimate, given 𝐹∈lip₀(𝑋,𝑑^𝛼)^∗∗ and 𝑥∈𝑋, notice that Δ⁻¹𝐶_𝜑Δ(𝐹)𝛿_𝑥=𝐶_𝜑Δ(𝐹)(𝑥)=𝐶_𝜑(Δ(𝐹𝑥)(𝐹𝜑(𝑥𝛿))(=Δ))=𝐹_𝜑(𝑥)=𝐹𝛿_𝑥∘𝐶_𝜑∣_{lip0(𝑋,𝑑𝛼)}=𝐹∘𝐶_𝜑∣_{lip0(𝑋,𝑑𝛼)}^∗𝛿_𝑥=𝐶_𝜑∣_{lip0(𝑋,𝑑𝛼)}^∗∗(𝐹)𝛿_𝑥. (5.5) Since lip₀(𝑋,𝑑^𝛼)^∗=span{𝛿_𝑥∶𝑥∈𝑋}, we conclude that Δ⁻¹𝐶_𝜑Δ=(𝐶_𝜑|_{lip0(𝑋,𝑑𝛼)})^∗∗. Finally, using the relationship between the essential norm of an operator and that of its second adjoint, and applying Theorem 4.1, we get 𝐶_𝜑𝑒=Δ⁻¹𝐶_𝜑Δ_𝑒=𝐶_𝜑∣_{lip0(𝑋,𝑑𝛼)}^∗∗_𝑒≤𝐶_𝜑∣_{lip0(𝑋,𝑑𝛼)𝑒}≤lim_𝑡→0sup_{0<𝑑𝑥,𝑦<𝑡}𝑑𝜑(𝑥),𝜑𝑦^𝛼𝑑𝑥,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 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.