# Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Authors: A. Jiménez-Vargas; Miguel Lacruz; Moisés Villegas-Vallecillos
Journal: Abstract and Applied Analysis
 (2011)

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## 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 Cφ induced by the symbol φ on the spaces lip0(X,dα) and Lip0(X,dα) is given by the formula ∥Cφ∥e=limt→0sup0<d(x,y)<t(d(φ(x),φ(y))α/d(x,y)α) whenever the dual space lip0(X,dα)∗ has the approximation property. This happens in particular when X is an infinite compact subset of a finite-dimensional normed linear space.


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## Body

## 1. Introduction

Let(X,d) be a compact metric space with a distinguished point e∈X and 0<α<1. The formula dα(x,y)=d(x,y)α defines a new metric on X, and the metric space (X,dα) is said to be a Hölder metric space of orderα. As usual, 𝕂 denotes the field of real or complex numbers.TheLipschitz spaceLip0(X,dα) is the Banach space of all Lipschitz functions f:X→𝕂 on the Hölder metric space (X,dα) for which f(e)=0 under the standard Lipschitz norm (1.1)Lα(f)=sup⁡{|f(x)-f(y)|d(x,y)α:x,y∈X,x≠y}.
Notice that the Lipschitz functions on (X,dα) are precisely the Hölder functions of orderα on (X,d).Thelittle Lipschitz spacelip0(X,dα) is the closed subspace consisting of those functions f∈Lip0(X,dα) that satisfy the following local flatness condition: (1.2)lim⁡t→0sup⁡0<d(x,y)<t|f(x)-f(y)|d(x,y)α=0.The Lipschitz spaceLip0(X,dα) has a canonical predual ℱ(X,dα), the free Lipschitz space on (X,dα), also known as the Arens-Eells space in [1], that can be defined as the closed linear span of the point evaluations(1.3)δx(f)=f(x),(x∈X,f∈Lip0(X,dα)),
in the dual space Lip0(X,dα)*. As it turns out, ℱ(X,dα) is itself the dual space of lip0(X,dα). 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 byℒ(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, (1.4)‖T‖e=inf⁡{‖T-K‖:K∈K(E)}.
It is clear that an operator T∈ℒ(E) is compact if and only if ∥T∥e=0.Recall that a Banach spaceE 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 (1.5)L(φ):=sup⁡{d(φ(x),φ(y))d(x,y):x,y∈X,x≠y}<∞.
The composition operatorCφ:lip0(X,dα)→lip0(X,dα) is defined by the expression (1.6)(Cφf)(x)=f(φ(x)),(x∈X,f∈lip0(X,dα)).The aim of this paper is to give lower and upper estimates for the essential norm of the composition operatorCφ on lip0(X,dα) 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 Cφ. We show that (1.7)‖Cφ‖=sup⁡x≠yd(φ(x),φ(y))αd(x,y)α.
The prototype of a formula as above with α=1 was provided by Weaver in [1, Proposition  1.8.2] for the composition operator Cφ on the space Lip0(X,d). In Section 3, we give a lower bound for the essential norm of the operator Cφ, namely, (1.8)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤‖Cφ‖e.
Section 4 contains our main result. When the dual space lip0(X,dα)* has the approximation property, we show that (1.9)‖Cφ‖e≤lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.
The proof of this inequality depends on some results involving shrinking compact approximating sequences on lip0(X,dα). Using the fact that the space Lip0(X,dα) is isometrically isomorphic to the second dual of lip0(X,dα), 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 Cφ on Lip0(X,dα).It is natural to ask for some examples where the dual spacelip0(X,dα)* has the approximation property. For instance, this happens if X is uniformly discrete, that is, infx≠yd(x,y)>0 (see [2, Proposition  4.4]). Also, lip0(X,dα)* has the approximation property whenever the space lip0(X,dα) is isomorphic to c0 and hence Lip0(X,dα) is isomorphic to ℓ∞ and ℱ(X,dα) is isomorphic to ℓ1.A classical result of Bonic et al. [7] ensures that lip0(X,dα) 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,dα) 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,dα) Lipschitz embeds in the euclidean space ℝn. Using this result, Kalton observed that if a compact metric space (X,d) satisfies the doubling condition, then the space lip0(X,dα) 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 ofCφ on Lip0(X,dα)

The aim of this section is to derive a formula for the norm of the composition operatorCφ on lip0(X,dα) in terms of the Lipschitz constant of φ. A similar expression was already provided by Weaver for the composition operator Cφ on the space Lip0(X,d), obtaining in [1, Proposition  1.8.2] the following identity: (2.1)‖Cφ‖=sup⁡x≠yd(φ(x),φ(y))d(x,y).Theorem 2.1.
LetX 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 Cφ:lip0(X,dα)→lip0(X,dα) is given by the expression
(2.2)‖Cφ‖=sup⁡x≠yd(φ(x),φ(y))αd(x,y)α.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 anyf∈lip0(X,dα) with Lα(f)≤1, we have
(2.3)Lα(Cφf)=sup⁡x≠y|f(φ(x))-f(φ(y))|d(x,y)α≤sup⁡φ(x)≠φ(y)|f(φ(x))-f(φ(y))|d(φ(x),φ(y))α⋅sup⁡x≠yd(φ(x),φ(y))αd(x,y)α≤Lα(f)⋅sup⁡x≠yd(φ(x),φ(y))αd(x,y)α≤sup⁡x≠yd(φ(x),φ(y))αd(x,y)α,
and so
(2.4)‖Cφ‖=sup⁡Lα(f)≤1Lα(Cφf)≤sup⁡x≠yd(φ(x),φ(y))αd(x,y)α.
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
(2.5)h(z)=d(z,φ(y))β-d(z,φ(x))β2d(φ(x),φ(y))β-α,
and f:X→ℝ by
(2.6)f(z)=h(z)-h(e).
It is not hard to show that f∈lip0(X,dα) with Lα(f)=1 (see, for instance, [9]), so
(2.7)‖Cφ‖≥Lα(Cφf)≥|f(φ(x))-f(φ(y))|d(x,y)α=d(φ(x),φ(y))αd(x,y)α.
Taking supremum over x and y, we conclude that
(2.8)‖Cφ‖≥sup⁡x≠yd(φ(x),φ(y))αd(x,y)α.


## 3. The Lower Estimate of the Essential Norm ofCφ on Lip0(X,dα)

Next we bound from below the essential norm ofCφ on lip0(X,dα) by means of an asymptotic quantity that measures the local flatness of φ.Theorem 3.1.
LetX 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 Cφ:lip0(X,dα)→lip0(X,dα) satisfies the lower estimate
(3.1)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤‖Cφ‖e.We will need the following description of the weak convergence inlip0(X,dα). This result is part of the folklore and it is immediate from the Banach-Steinhaus theorem since lip0(X,dα)*=span¯{δx:x∈X} (see Weaver [1, Theorem  3.3.3]).Lemma 3.2.
LetX be a pointed compact metric space, 0<α<1 and {fn} a sequence in lip0(X,dα). Then {fn} converges to 0 weakly in lip0(X,dα) if and only if {fn} is bounded in lip0(X,dα) and converges to 0 pointwise on X.Proof of Theorem3.1.
Since the mappingφ:X→X is Lipschitz, the function
(3.2)t⟼sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α,(t>0)
is well defined. It is easy to check that
(3.3)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α=inf⁡t>0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.
Now, for every natural number n we can take a real number tn such that 0<tn<[2/n(1+L(φ)α)]1/α and two points xn,yn∈X such that 0<d(xn,yn)<tn, satisfying
(3.4)sup⁡0<d(x,y)<tnd(φ(x),φ(y))αd(x,y)α<inf⁡t>0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α+1n,sup⁡0<d(x,y)<tnd(φ(x),φ(y))αd(x,y)α-1n<d(φ(xn),φ(yn))αd(xn,yn)α.
In this way we obtain two sequences {xn} and {yn} in X such that
(3.5)0<d(xn,yn)<[2n(1+L(φ)α)]1/α,|d(φ(xn),φ(yn))αd(xn,yn)α-inf⁡t>0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α|<1n,
for all n∈ℕ, and this last inequality implies that
(3.6)inf⁡t>0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α=lim⁡n→∞d(φ(xn),φ(yn))αd(xn,yn)α.
Now take xn,yn∈X for which φ(xn)≠φ(yn) and choose β∈]α,1[. Define hn,fn:X→ℝ by
(3.7)hn(x)=d(x,φ(yn))β-d(x,φ(xn))β2d(φ(xn),φ(yn))β-α,fn(x)=hn(x)-hn(e).
Using (3.5), we have ∥hn∥∞≤1/n. Clearly, fn∈lip0(X,dα) with Lα(fn)=Lα(hn)=1 and ∥fn∥∞≤2/n for all n∈ℕ. Moreover, an easy calculation shows that |fn(φ(xn))-fn(φ(yn))|=d(φ(xn),φ(yn))α. Since
(3.8)d(φ(xn),φ(yn))αd(xn,yn)α=|fn(φ(xn))-fn(φ(yn))|d(xn,yn)α≤Lα(Cφfn),
for all n∈ℕ, we have
(3.9)lim⁡n→∞d(φ(xn),φ(yn))αd(xn,yn)α≤limsup⁡n→∞Lα(Cφfn).
By Lemma 3.2, {fn}→0 weakly in lip0(X,dα). Thus, if K is any compact operator from lip0(X,dα)into lip0(X,dα), then limn→∞Lα(Kfn)=0 because compact operators map weakly convergent sequences into norm convergent sequences. It follows that
(3.10)limsup⁡n→∞Lα(Cφfn)=limsup⁡n→∞(Lα(Cφfn)-Lα(Kfn))≤limsup⁡n→∞Lα((Cφ-K)fn)≤‖Cφ-K‖.
Combining (3.3), (3.6), (3.9), and (3.10), we conclude that
(3.11)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤‖Cφ-K‖.
By taking the infimum on both sides of this inequality over all compact operators K on lip0(X,dα), we obtain the lower estimate
(3.12)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤‖Cφ‖e.


## 4. The Upper Estimate of the Essential Norm ofCφ on Lip0(X,dα)

Now we prove that the lower bound of the essential norm ofCφ on lip0(X,dα) obtained in Section 3 is also an upper bound whenever the dual space lip0(X,dα)* has the approximation property.Theorem 4.1.
LetX be a pointed compact metric space and 0<α<1. Suppose that the dual space lip0(X,dα)* has the approximation property. Let φ:X→X be a base point preserving Lipschitz mapping. Then the essential norm of the composition operator Cφ:lip0(X,dα)→lip0(X,dα) satisfies
(4.1)‖Cφ‖e≤lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.The strategy for the proof of Theorem4.1 is to work with a sequence {Kn} of compact operators on lip0(X,dα) 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{Kn} is called a compact approximating sequence for a separable Banach space E if each Kn:E→E is a compact operator and limn→∞∥(I-Kn)f∥=0 for every f∈E, where I denotes the identity operator on E. Also, we say that {Kn} is shrinking if limn→∞∥(I-Kn)*f*∥=0 for every f*∈E*.Johnson [10, Theorem  2] showed that both the Banach space lip0(X,dα) and its dual space lip0(X,dα)* are separable.On the other hand, theBanach-Mazur distance between isomorphic Banach spaces E,F is defined by (4.2)d(E,F)=inf⁡{‖T‖⋅‖T-1‖:TisanisomorphismofEontoF}.
We say that Eembeds almost isometrically intoF provided that for every ε>0, there is a subspace Fε⊂F such that d(E,Fε)<1+ε.The next proposition is immediate from a result of Kalton [11].Proposition 4.2 (see [11, Corollary  3]).
Let{Kn} be a sequence of compact operators between Banach spaces E and F and let us suppose that limn→∞〈Kn*f*,f**〉=0 for all f*∈F* and f**∈E**. Then there exists a sequence {Knc} of compact operators such that Knc∈conv{Km:m≥n} and limn→∞∥Knc∥=0.Lemma 4.3.
LetX be a pointed compact metric space and 0<α<1. Suppose that the dual space lip0(X,dα)* has the approximation property. Then there is a shrinking compact approximating sequence {Kn} on lip0(X,dα) such that limsupn→∞∥I-Kn∥≤1.Proof.
Since the dual spacelip0(X,dα)* is separable and has the approximation property, it has the metric approximation property and therefore there is a shrinking compact approximating sequence {Sn} on lip0(X,dα). We claim that for every j∈ℕ, there exist a natural nj≥j and a compact operator Kj on lip0(X,dα) in the convex hull of the set {Sm:m≥nj} such that ∥I-Kj∥<(1+1/j)2.
Fixj∈ℕ. Now, for the proof of our claim, we use a result that ensures that lip0(X,dα) 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 Fj⊂c0 and an isomorphism Tj:lip0(X,dα)→Fj such that ∥Tj∥·∥Tj-1∥<1+(1/j). Now consider Mn:=TjSnTj-1, and notice that {Mn} is a shrinking compact approximating sequence on Fj. Next, let {Pn} be the sequence of projections on c0 defined by
(4.3)(Pnx)(k)=x(k),(1≤k≤n),(Pnx)(k)=0,(k>n),
for all x∈c0, and let J:Fj↪c0 be the inclusion map. Then consider the sequence of compact operators Dn:=PnJ-JMn defined from Fj into c0. Notice that limn→∞〈Dn*f*,f**〉=0 for all f*∈c0* and f**∈Fj**. It follows from Proposition 4.2 that there is a sequence of operators {Dnc} such that Dnc∈conv{Dm:m≥n} and limn→∞∥Dnc∥=0. This gives rise to a pair of shrinking compact approximating sequences {Pnc} and {Mnc} such that, for each n∈ℕ,Pnc∈conv{Pm:m≥n}, Mnc∈conv{Mm:m≥n}, and limn→∞∥PncJ-JMnc∥=0. Now, consider Ln:=Tj-1MncTj. We have
(4.4)‖I-Ln‖=‖I-Tj-1MncTj‖≤‖Tj-1‖⋅‖Tj‖⋅‖I-Mnc‖≤(1+1j)⋅‖I-Mnc‖=(1+1j)⋅‖J(I-Mnc)‖≤(1+1j)⋅(‖(I-Pnc)J‖+‖PncJ-JMnc‖)≤(1+1j)⋅(1+‖PncJ-JMnc‖),
for all n∈ℕ. Finally, choose nj≥j large enough so that ∥PnjcJ-JMnjc∥<1/j and conclude that ∥I-Lnj∥<(1+(1/j))2. The claim is proved if we take Kj:=Lnj.
The proof of the lemma will be finished if we show that{Kj} is a shrinking compact approximating sequence on lip0(X,dα). Let f∈lip0(X,dα) and ε>0. Since limn→∞Lα(f-Snf)=0, there exists m0∈ℕ such that Lα(f-Snf)<ε for n≥m0. If j≥m0, using that Kj∈conv{Sm:m≥nj} and nj≥j, we conclude that Lα(f-Kjf)<ε. Hence limj→∞Lα(f-Kjf)=0. This shows that {Kj} is approximating on lip0(X,dα) and similarly it is seen that {Kj} is shrinking.There is another preliminary result that is needed for the proof of Theorem4.1 and that can be stated as follows.Lemma 4.4.
LetX be a pointed compact metric space, 0<α<1 and φ:X→X a base point preserving Lipschitz mapping. Let {Kn} be a shrinking compact approximating sequence on lip0(X,dα).
Then, for eacht>0,
(4.5)lim⁡n→∞sup⁡Lα(f)≤1sup⁡d(x,y)≥t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α=0.Proof.
Fixt>0. Since the inequality ∥f∥∞≤diam(X)α·Lα(f) is satisfied for all f∈lip0(X,dα), there is a continuous injection J:(lip0(X,dα),Lα(·))↪(lip0(X,dα),∥·∥∞). 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.
LetB be the unit ball of (lip0(X,dα),∥·∥∞)*. Since the bounded sequence of operators {(I-Kn)*} converges to zero pointwise on lip0(X,dα)* and J*(B) is a relatively compact set in (lip0(X,dα),Lα(·))*, it follows that limn→∞∥(I-Kn)*J*∥=0. Thus, limn→∞∥J(I-Kn)∥=0. Let ε>0 be given and choose m∈ℕ such that if n≥m, then ∥(I-Kn)f∥∞<εtα/4 for all f∈lip0(X,dα) with Lα(f)≤1. For n≥m, we have
(4.6)|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α≤2‖(I-Kn)f‖∞tα<ε2,
whenever f∈lip0(X,dα) with Lα(f)≤1 and x,y∈X such that d(x,y)≥t. Finally, we get
(4.7)sup⁡Lα(f)≤1sup⁡d(x,y)≥t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α<ε,
as required.We now are ready to prove our main result.Proof of Theorem4.1.
Let{Kn} be the sequence of operators on lip0(X,dα) provided by Lemma 4.3. Since each Kn is a compact operator, so is the product CφKn:lip0(X,dα)→lip0(X,dα) and therefore
(4.8)‖Cφ‖e≤‖Cφ(I-Kn)‖.
Next, fix t>0 and notice that
(4.9)‖Cφ(I-Kn)‖=sup⁡Lα(f)≤1Lα(Cφ(I-Kn)f)=sup⁡Lα(f)≤1sup⁡x≠y|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α≤sup⁡Lα(f)≤1sup⁡0<d(x,y)<t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α+sup⁡Lα(f)≤1sup⁡d(x,y)≥t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α.
Then, for every f∈lip0(X,dα), we have
(4.10)sup⁡0<d(x,y)<t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α≤sup⁡φ(x)≠φ(y)|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(φ(x),φ(y))α⋅sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤Lα[(I-Kn)f]⋅sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α,
so that
(4.11)sup⁡Lα(f)≤1sup⁡0<d(x,y)<t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α≤‖I-Kn‖sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.
Now, combining the above inequalities, we obtain
(4.12)‖Cφ‖e≤‖I-Kn‖sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α+sup⁡Lα(f)≤1sup⁡d(x,y)≥t|[(I-Kn)f](φ(x))-[(I-Kn)f](φ(y))|d(x,y)α.
Letting n→∞ and using Lemmas 4.3 and 4.4, we conclude that
(4.13)‖Cφ‖e≤sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.
Finally, taking limits as t→0 yields the desired inequality.


## 5. The Essential Norm ofCφ on Lip0(X,dα)

Now we extend the estimates on the essential norm of a composition operator to the spacesLip0(X,dα). Recall that lip0(X,dα)** is isometrically isomorphic to Lip0(X,dα) 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,dα)**→Lip0(X,dα), defined by (5.1)Δ(F)(x)=F(δx),(F∈lip0(X,dα)**,x∈X),
is an isometric isomorphism.IfT 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.
LetX 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 Cφ:Lip0(X,dα)→Lip0(X,dα) satisfies the lower estimate
(5.2)‖Cφ‖e≥lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.
If, in addition, the space lip0(X,dα)* has the approximation property, then one has the upper estimate
(5.3)‖Cφ‖e≤lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α.Proof.
Let us start with the lower estimate. Let{fn} be the weakly null sequence in lip0(X,dα) that we constructed for the proof of Theorem 3.1. Then the sequence {fn} is weakly null in Lip0(X,dα). Thus, if K is any compact operator on Lip0(X,dα), we have limn→∞Lα(Kfn)=0. Hence, the same computation we performed in Theorem 3.1 yields the lower estimate
(5.4)lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α≤‖Cφ‖e.
Now, for the upper estimate, given F∈lip0(X,dα)** and x∈X, notice that
(5.5)(Δ-1CφΔ)(F)(δx)=((CφΔ)(F))(x)=Cφ(Δ(F))(x)=Δ(F)(φ(x))=F(δφ(x))=F(δx∘Cφ∣lip0(X,dα))=(F∘(Cφ∣lip0(X,dα))*)(δx)=(Cφ∣lip0(X,dα))**(F)(δx).
Since lip0(X,dα)*=span¯{δx:x∈X}, we conclude that Δ-1CφΔ=(Cφ|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
(5.6)‖Cφ‖e=‖Δ-1CφΔ‖e=‖(Cφ∣lip0(X,dα))**‖e≤‖Cφ∣lip0(X,dα)‖e≤lim⁡t→0sup⁡0<d(x,y)<td(φ(x),φ(y))αd(x,y)α,
as we wanted.



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*Source: 590853-2011-11-28.xml*