Volume 2013, Issue 1 741050

Research Article

Open Access

Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

Dongyang Chen,

Dongyang Chen

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China xmu.edu.cn

Search for more papers by this author

Lei Li,

Corresponding Author

Lei Li

[email protected]

orcid.org/0000-0003-1177-7539

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China nankai.edu.cn

Search for more papers by this author

Risheng Wang,

Risheng Wang

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China nankai.edu.cn

Search for more papers by this author

Ya-Shu Wang,

Ya-Shu Wang

Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li 32023, Taiwan cycu.edu.tw

Search for more papers by this author

Dongyang Chen,

Dongyang Chen

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China xmu.edu.cn

Search for more papers by this author

Lei Li,

Corresponding Author

Lei Li

[email protected]

orcid.org/0000-0003-1177-7539

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China nankai.edu.cn

Search for more papers by this author

Risheng Wang,

Risheng Wang

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China nankai.edu.cn

Search for more papers by this author

Ya-Shu Wang,

Ya-Shu Wang

Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li 32023, Taiwan cycu.edu.tw

Search for more papers by this author

First published: 30 October 2013

https://doi.org/10.1155/2013/741050

Citations: 1

Academic Editor: Fabio M. Camilli

Share a link

Email
Wechat
Bluesky

Abstract

We will give the α-Lipschitz version of the Banach-Stone type theorems for lattice-valued α-Lipschitz functions on some metric spaces. In particular, when X and Y are bounded metric spaces, if T : Lip(X) → Lip(Y) is a nonvanishing preserver, then T is a weighted composition operator Tf = h · f∘φ, where φ : Y → X is a Lipschitz homeomorphism. We also characterize the compact weighted composition operators between spaces of Lipschitz functions.

1. Introduction

The classical Banach-Stone theorem tells us that, when X and Y are compact Hausdorff spaces, every linear surjective isometry from C(X) onto C(Y) can be written as a weighted composition operator; that is, it is of the form

()

where φ is a homeomorphism from Y onto X and J ∈ C(Y) with |J(y)| = 1 for all y ∈ Y. The theorem has many variable extensions concerning isometries, algebra isomorphisms, and disjointness preserving mappings between continuous function spaces; and we refer the surveys [1, 2] for more history about Banach-Stone theorems. Moreover, Kamowitz [3] gave the representation and spectrum of the compact weighted composition operators on the continuous functions.

Cao et al. stated a lattice version of the classical Banach-Stone theorem in [4]. Later, Chen et al. [5], Ercan and Önal [6, 7], and Miao et al. [8] generalized this result. When X, Y are compact Hausdorff spaces and E, F are Banach lattices, by the main results of [5, 7], we can see that every vector lattice isomorphism T from C(X, E) onto C(Y, F) preserving the nonvanishing functions must be a weighted composition operator.

Garrido and Jaramillo [9, 10] and Weaver [11] tackled the Banach-Stone type theorem for lattices of real Lipschitz functions. Later, Jiménez-Vargas and Villegas-Vallecillos [12] proved that two little Lipschitz algebras are order isomorphic if and only if the corresponding compact metric spaces are Lipschitz homeomorphic. Recently, Jiménez-Vargas et al. [13] presented a Lipschitz version of the result in [5], in which the underlying spaces should be compact.

Our first goal of this paper is to prove the Banach-Stone type theorem in the setting of lattice-valued α-Lipschitz functions. Section 2 is devoted to the preliminaries about vector lattices and α-Lipschitz functions. Then we will give the α-Lipschitz version of Banach-Stone theorem in Section 3. In particular, when X, Y are bounded metric spaces, if T : Lip(X) → Lip(Y) is a nonvanishing preserver, then we will show that T is a weighted composition operator Tf = h · f∘φ, where φ : Y → X is a Lipschitz homeomorphism. Our second aim is to give the characterization of compact weighted composition operators on the α-Lipschitz functions.

2. Preliminaries

An ordered vector space E is said to be a vector lattice if max{x, y} exists for any x, y in E. A vector lattice E is said to be a Banach lattice if it is complete under its norm ∥·∥ and satisfies the Riesz law:

()

where |x | = max{x, −x}.

Let (X, d) be a metric space and E a Banach space; if 0 < α ≤ 1, a function f from X to E is said to be α-Lipschitz if

()

The α-Lipschitz function space Lip_α(X, E) is the space of all E-valued α-Lipschitz functions on X.

is the Banach space of all bounded α-Lipschitz functions f : X → E with the α-Lipschitz norm

()

where ∥f∥_∞ = sup{∥f(x)∥ : x ∈ X}. Furthermore, the little Lipschitz space lip_α(X, E) is then defined to be the closed subspace of Lip_α(X, E) of these functions f with the following property: for every ε > 0, there exists δ > 0 such that ∥f(x₁) − f(x₂)∥ < εd^α(x₁, x₂) whenever d(x₁, x₂) ≤ δ.

is the subspace of lip_α(X, E) consisting of all bounded functions. Notice that when α = 1, Lip_α(X, E) is just Lipschitz space Lip(X, E). Moreover, if E is a Banach lattice, then Lip_α(X, E) is a vector lattice with the usual pointwise order

()

However,

is not a Banach lattice since ∥·∥ does not satisfy the Riesz law in general.

A mapping φ from Y to X is said to be a α-Lipschitz homeomorphism if it is bijective and φ and φ⁻¹ are both α-Lipschitz. If f is in Lip_α(X) and e is a vector in E, denote by f ⊗ e the function x ↦ f(x)e in Lip_α(X, E). In particular, 1 ⊗ e denotes the constant function x ↦ e on X. For any function f in Lip_α(X, E), the zero set {x ∈ X : f(x) = 0} of f is denoted by z(f) and its cozero set {x ∈ X : f(x) ≠ 0} is coz(f), and f is said to be nonvanishing if z(f) = ∅. An operator T : Lip_α(X, E) → Lip_α(Y, F) is said to be a nonvanishing preserver if

()

T is said to be a Riesz isomorphism if T(f∨g) = Tf∨Tg and T(f∧g) = Tf∧Tg for any f, g ∈ Lip_α(X, E).

3. Nonvanishing Preservers on Lipschitz Functions

In this section, our results will be valid (with the same proof) for different kinds of spaces. For this reason we first consider several situations to work in. Throughout this section we will assume that 0 < α ≤ 1, X, Y are metric spaces and E, F are Banach lattices.

Context 1. , .

Context 2 (0 < α < 1). , .

This means that when we refer to X, Y, A(X, E), A(Y, F), we assume that all of them are included at the same time in one of the above two contexts.

Suppose that X is a metric space and 0 < α ≤ 1, for any x₁ ≠ x₂ ∈ X, the function

()

belongs to

. Moreover, if 0 < α < 1, then we can find β > 0 such that α < β < 1, and the function

()

belongs to

. The function f defined in (7) or (8) has the property: 0 ≤ f ≤ 1, f(x₁) = 0, f(x₂) = 1 and z(f) = {x ∈ X : d(x, x₂) ≥ d(x₁, x₂)}.

Theorem 1. Let T : A(X, E) → A(Y, F) be a Riesz isomorphism preserving nonvanishing functions. Then T carries the form

()

Here, φ is a homeomorphism from Y onto X and all fiber linear maps Jy : E → F are isomorphisms.

Remark 2. When α = 1, the previous theorem is not valid for the little Lipschitz space lip₁(X, E), where X is a connected Banach and E is a Banach lattice. Note that if X is a connected Banach spaces, we have that consisting of all E-valued constant functions defined on X. Let φ be any map from ℝ² to ℝ and T : lip₁(ℝ, E) → lip₁(ℝ², E) a linear bijection operator defined by

()

It is obvious that the operator T is a Riesz isomorphism preserving nonvanishing functions with a weighted composition representation, but ℝ and ℝ² are not homeomorphic.

It is easy to prove the following lemma.

Lemma 3. T preserves common zeros, that is,

()

for any f₁, …, f_n ∈ A(X, E) and n ∈ ℕ.

Proof of Theorem 1. In the above contexts, A(X, E) and A(Y, F) contain constant functions, so E_y = E and F_y = F, where E_y, F_y are defined in [14, Definition 3.8]. Therefore, by [14, Theorem 3.1] we can derive the result.

Lemma 4. In the Contexts 1 and 2, T is automatically continuous.

Proof. We are going to use the Closed Graph Theorem to prove this lemma. Suppose that the sequence of functions {f_n} converges to f₀ in A(X, E) and {Tf_n} converges to g₀ in A(Y, F); then for any x ∈ X and y ∈ Y, we have that {f_n(x)} converges to f₀(x) in E and {(Tf_n)(y)} converges to g₀(y) in F, respectively. Notice that, for any x ∈ X, Jφ⁻¹(x) : E → F is continuous; then one can derive that

()

for all x in X. Since φ is a bijection from Y onto X, we get that the sequence {(Tf_n)(y)} converges to (Tf₀)(y) for all y in Y, and hence g₀ = Tf₀. This means that T is a closed operator from A(X, E) to A(Y, F), and then T is continuous.

In order to prove that φ is a α-Lipschitz map from Y onto X, we need the following lemma, and some idea of the proof comes from [15, Lemma 5.8].

Lemma 5. For any fixed element e ∈ E with ∥e∥ = 1, we have that

()

Proof. By Theorem 1 we can also find a map from X to Iso(F, E) (which is the set of all linear isomorphisms from F to E) and a bijection from X onto Y such that

()

for all x ∈ X and g ∈ A(Y, F). From the definition of ψ, φ, and

, we can see that

Suppose on the contrary that there exists a sequence {y_n} ⊂ Y such that ∥T(1 ⊗ e)(y_n)∥ = ∥(Jy_n)(e)∥ ≤ 2⁻²ⁿ for all n ∈ ℕ. If {y_n} has a limit point y^′ in Y, notice that T preserves nonvanishing functions, then we can see that (Jy^′)(e) = 0 and hence e = 0. This leads to a contradiction. On the other hand, if there exists a positive scalar τ > 0 such that d^α(y_n, y_m) ≥ τ for any n, m ∈ ℕ with n ≠ m, when we take the norm one element

()

then we can derive that

()

for all n ∈ ℕ. Therefore, for any n ∈ ℕ, we know that

()

Moreover, for any n ∈ ℕ, by the similar manner of (7) and (8) we can define the function ψ_n(y) ∈ A^b(Y) such that 0 ≤ ψ_n ≤ 1, L_α(ψ_n) ≤ m for some m > 0, ψ_n(y_n) = 1 and ψ_n(y) = 0 for all y such that d(y, y_n) ≥ τ/2. When put

()

we can see that h₀ belongs to A^b(Y, F) and h₀(y_n) = b_n/2ⁿ for n > 1. Then one can conclude that

()

and hence

()

This is a contradiction in Contexts 1 and 2 since ∥T⁻¹h₀∥_∞ < ∞.

Theorem 6. Suppose that X, Y are bounded metric spaces in the Contexts 1 and 2; then φ is a α-Lipschitz map from Y onto X.

Proof. We can define the linear map from A(X) to A(Y) by

()

We have to show that

is well defined at first. For any fixed element e ∈ E with ∥e∥ = 1, from Lemma 5 we can choose a positive scalar ν such that ∥T(1 ⊗ e)(y)∥ ≥ ν > 0 for all y in Y, and then it is easy to see that the function h which maps y to 1/∥T(1 ⊗ e)(y)∥ belongs to A(Y).

Assume that f is a positive function in A(X); one can get that, for any y₁, y₂ ∈ Y,

()

that is,

is a bounded α-Lipschitz function. Moreover, in Context 2 we can derive that

is also a little Lipschitz function. This means that the function

belongs to A(Y). Therefore,

is a well-defined bijective linear operator from A(X) onto A(Y), and

is also a nonvanishing preserver.

Suppose that {f_n} is a sequence which converges to 0 in A(X) and the sequence converges to g₀ in A(Y). For any n ∈ ℕ and y ∈ Y, , and hence we have that {f_n(φ(y))} converges to g₀(y) for all y ∈ Y. Notice that {f_n} converges to 0; one can conclude that {f_n(x)} converges to 0 for all x ∈ X, and, since φ is a bijective map from Y onto X, we have that g₀(y) = 0 for any y in Y. Therefore, is a closed operator and hence is continuous.

For any y₁ and y₂ in Y, there exists a function f₀ ∈ A(X) such that ∥f₀∥ ≤ D(X) + D(X) ^1−α and f₀(φ(y₁)) = d(φ(y₁), φ(y₂)) and f₀(φ(y₂)) = 0 (in fact, f₀(x) = d(x, φ(y₂)) has the properties that we need). Here D(X) denotes the diameter of X. Then we can derive that

()

Furthermore, we have that

()

and this means that φ is a α-Lipschitz mapping from Y onto X. Similarly, we can see that φ⁻¹ is also α-Lipschitz, and then φ is a α-Lipschitz homeomorphism.

For the spaces of scalar-valued Lipschitz functions, we give a complete characterization of nonvanishing preservers. But at first we need to recall a special case of [16, Lemma 25].

Lemma 7. Let A(X), A(Y) be in Contexts 1 and 2. Suppose that T : A(X) → A(Y) is a linear nonvanishing preserver; then the map S : A(X) → A(Y) given by

()

is a Riesz isomorphism preserving nonvanishing functions.

Proof. For completeness, we will sketch the proof. Observe that T1 is never vanishing. If f ∈ A(X) and λ ∈ ℝ, then λ ∈ range f if and only if 0 ∈ range(f − λ) if and only if 0 ∈ range(Tf − λT1) if and only if λ ∈ range Tf/T1. In particular, if f ≥ 0, then Tf/T1 ≥ 0. Let Y₊ = {y ∈ Y : (T1)(y) > 0} and Y₋ = {y ∈ Y : (T1)(y) < 0}. Then Y₊ ∪ Y₋ is a partition of Y into two open sets.

Suppose that f ∈ A(X) and f ≥ 0. Then Tf ≥ 0 on Y₊ and Tf ≤ 0 on Y₋. Hence Tf · T1/|T1| = |Tf | ∈ A(Y). For any f ∈ A(X), we have that f₊, f₋ ∈ A(X), and |T(f₊)| = T(f₊) · T1/|T1| and |T(f₋)| = T(f₋) · T1/|T1|. Then we can derive that

()

This means that S is well defined. Moreover, it is easy to check that S is bijective.

From the previous paragraph, if 0 ≤ f ∈ A(X), then Sf = |Tf | ≥ 0. If f ∈ A(X) and g = Sf ≥ 0, then by the above,

()

By [17, Lemma 2.3], T is biseparating, and hence T(f₊) · T(f₋) = 0. It follows that T(f₋) = 0 and thus f₋ = 0. Therefore, f ≥ 0. Thus S is a Riesz isomorphism. It is trivial to check that 0 ∈ range f if 0 ∈ range Sf for any f ∈ A(X).

Theorem 8. Suppose that X, Y are bounded metric spaces and T is a nonvanishing preserver between the following function spaces:

(i)
0 < α ≤ 1 and T : Lip_α(X) → Lip_α(Y);
(ii)
0 < α < 1 and T : Lip_α(X) → Lip_α(Y).

Then T is a weighted composition operator of the form

()

Here h = T1 and φ : Y → X is a α-Lipschitz map.

Proof. By Lemma 7 we have that T is a Riesz isomorphism. Then by Theorem 6 we can derive the conclusion.

In Theorem 8, the boundedness of the metric spaces can not be dropped.

Example 9. Let ℕ₁ be the positive integers with the discrete metric, and we can derive that ℕ₁ is not Lipchitz homeomorphic to ℕ. By [18, Example 1.6.4] we can derive that Lip^b(ℕ) = Lip^b(ℕ₁) = ℓ^∞, and then the identity map I : Lip^b(ℕ) → Lip^b(ℕ₁) is a nonvanishing preserver. However, the underlying metric spaces are not Lipschitz homeomorphic.

4. Compact Weighted Composition Operators on Lipschitz Spaces

Suppose that X, Y are metric spaces, 0 < α ≤ 1, and

is a weighted composition operator, that is,

()

Here h = T1 and φ : Y → X is a α-Lipschitz mapping. Put Y₀ = {y ∈ Y : h(y) = 0}. Recall that φ : Y → X is supercontractive on Y^′ ⊂ Y if for each ε > 0 there exists δ > 0 such that d(φ(y₁), φ(y₂)) < εd(y₁, y₂) whenever y₁, y₂ ∈ Y^′ and 0 < d(y₁, y₂) < δ. In this section, we will characterize the compact weighted composition operator T and consider its spectrum.

Theorem 10. Suppose that T is compact. For any y₀ ∈ Y∖Y₀, there is an open neighborhood U₀ of y₀ such that φ is supercontractive on U₀ and φ(U₀) is totally bounded.

Proof. Since h(y₀) ≠ 0, we can find an open neighborhood U₀ of y₀ such that |h(y)|≥|h(y₀)|/2 > 0 for all y ∈ U₀. Suppose on the contrary that there exist {x_n}, {y_n} ⊂ U₀ such that d(x_n, y_n) → 0 and

()

for some ε₀ > 0. Without loss of generality we can assume that

Let

()

we can derive that

and |f_n(x₁) − f_n(x₂)| ≤ d^α(x₁, x₂) for any x₁, x₂ ∈ X. This implies that {f_n} is a bounded sequence in

. If T is compact, then there exists a subsequence

such that

. Since f_n → 0 uniformly, for any y ∈ Y, we have that

()

and then g₀ = 0. This means that

On the other hand, for any n ∈ ℕ, by the Mean Value Theorem we have that

()

Here

. Therefore, we can derive that L_α(Tf_n)↛0, and this is a contradiction.

On the other hand, suppose on the contrary that φ(U₀) is not totally bounded, then there exist a constant τ > 0 and z_n = φ(u_n) ∈ φ(U₀) such that d^α(z_n, z_m) > τ whenever n ≠ m. Let

()

then it is easy to see that f_n(z_n) = 0 and ∥f_n∥ ≤ 1. Moreover, for any n ≠ m, we can derive that

()

Therefore, {Tf_n} has no Cauchy subsequence, and hence T is not compact. This leads to a contradiction.

Theorem 11. Suppose that φ is supercontractive on Y∖Y₀ and φ(Y∖Y₀) is totally bounded; then the weighted composition operator defined by (29) is compact.

Proof. Let be a bounded sequence, that is, ∥f_n∥ ≤ M for some M > 0. Since φ(Y∖Y₀) is totally bounded, there exists a subsequence of {f_n}, which is also denoted by {f_n}, such that {f_n} is convergent uniformly in φ(Y∖Y₀). Denote the limit by f₀(x) for all x ∈ φ(Y∖Y₀). It is easy to verify that f₀ is a bounded Lipschitz function in φ(Y∖Y₀). By the similar argument of [18, Theorem 1.5.6] we can extend f₀ to be a bounded Lipschitz function in , which is also denoted by f₀. It suffices to show that {Tf_n} converges to Tf₀ in .

Since T is a weighted composition operator, it is easy to see that {Tf_n} converges to Tf₀ uniformly on Y. Let ε > 0 be given. Since φ is supercontractive on Y∖Y₀, there exists δ > 0 such that

()

whenever y₁, y₂ ∈ Y∖Y₀ and 0 < d(y₁, y₂) < δ.

We will show that L_α(Tf_n − Tf₀) → 0 by dividing into four cases as the following arguments. For any y₁, y₂ ∈ Y with y₁ ≠ y₂.

Case 1. If y₁, y₂ ∈ Y₀, we have that (Tf_n)(y_i) = (Tf₀)(y_i) = 0 for i = 1,2.

Case 2. If y₁, y₂ ∈ Y∖Y₀ and 0 < d(y₁, y₂) < δ, we have that

()

Moreover, by (36) we can derive that

()

Case 3. If y₁, y₂ ∈ Y∖Y₀ and d(y₁, y₂) > δ, we have that

()

Case 4. If y₁ ∈ Y∖Y₀ and y₂ ∈ Y₀, we have that h(y₂) = 0 and then

()

Hence we derive that L_α(Tf_n − Tf₀) → 0 and then Tf_n → Tf₀. This means that T is a compact operator.

By the similar argument, one can conclude the following results for the scalar-valued little Lipschitz function spaces.

Theorem 12. Let α ∈ (0,1). Suppose that is a nonzero weighted composition operator of the form (29).

(1)
If T is compact, then, for any y₀ ∈ Y∖Y₀, there is an open neighborhood U₀ of y₀ such that φ is supercontractive on U₀ and φ(U₀) is totally bounded.
(2)
If φ is supercontractive on Y∖Y₀ and φ(Y∖Y₀) is totally bounded, then T is compact.

Also here, the result of [19] also refers to the case where T is a composition operator.

Corollary 13. Suppose that X, Y are compact metric spaces, and T is a weighted composition operator of the form (29) between the following function spaces:

(i)
0 < α ≤ 1 and ;
(ii)
0 < α < 1 and .

Then T is compact if and only if φ is supercontractive on Y∖Y₀.

When T is a composition operator, that is, h = T1 = 1 in the form (29), then Y₀ = ∅ and we can establish the following results in [20, Theorem 1.1].

Corollary 14. Suppose that X, Y are metric spaces and is a composition operator; then T is compact if and only if φ is supercontractive and φ(Y) is totally bounded.

In the following part of this section we have X = Y. Define φ₀(x) = x and φ_n(x) = φ(φ_n−1(x)) for all x ∈ X by induction. A point x₀ ∈ X is said to be the fixed point of φ of order n, n ∈ ℕ, if φ_n(x₀) = x₀ and φ_i(x₀) ≠ x₀ for any i = 0,1, …, n − 1.

Theorem 15. Let X be a complete metric space and T : Lip^b(X) → Lip^b(X) a weighted composition operator of form (29) satisfying: φ is supercontractive on X∖X₀ and φ(X∖X₀) is totally bounded. Then we can derive that σ(T) = {0} ∪ 𝒮, where

()

Proof. Suppose that x₀ is a fixed point of φ of order n. If h(φ_k(x₀)) = 0 for some k, we can see that T is not surjective and hence 0 ∈ σ(T).

Assume that h(φ_k(x₀)) ≠ 0 for any k = 0,1, 2, …, n − 1 and λⁿ = h(x₀) ⋯ h(φ_n−1(x₀)).

When n = 1, we have that λ = h(x₀) and φ(x₀) = x₀. There exists g ∈ Lip^b(X) such that g(x₀) = 1. There is no f ∈ Lip^b(X) such that (λ − T)f = g. Indeed, if such f exists, we can derive that

()

and this is impossible. This means that λ ∈ σ(T).

When n ≥ 2, let δ : = min{d(φ_i(x₀), φ_j(x₀)) : 0 ≤ i ≠ j ≤ n − 1}, and define

()

Here h(φ₋₁(x₀)): = 1. Then, similar to the argument of [3, Proposition 3], we can derive that λ ∈ σ(T).

On the other hand, for each f ∈ Lip^b(X) with λf = Tf, for some λ ∉ {0} ∪ 𝒮, we will prove that f = 0. This implies that λ ∉ σ(T) and completes the proof.

From the assumption λf = Tf = h · f∘φ, for all x ∈ X and n ∈ ℕ, we derive that

()

Given any z ∈ X, let ℱ = {φ_n(z) : n ∈ ℕ ∪ {0}} and 𝒩 = {n ∈ ℕ:|h(φ_n(z))| ≥ δ₀}; here δ₀ is any fixed number with 0 < δ₀<|λ|. We provide that f(z) = 0, which implies that f = 0, by dividing into the following cases.

Case I (ℱ∩X₀ ≠ ∅). If there exists i₀ such that , by (44) we can see that

()

This implies that f(z) = 0.

Case II (ℱ ⊂ X∖X₀ and ℱ is finite). Let . Then there exists 0 ≤ k ≤ n₀ such that . This means that φ_k(z) is a fixed point of φ of order n₀ − k + 1. By (44), we have that

()

and f(φ_k(z)) = 0 since λ ∉ {0} ∪ 𝒮. Once again, by (44), we derive that

()

and f(z) = 0.

Case III (ℱ ⊂ X∖X₀, ℱ is infinite and 𝒩 is infinite). Notice that {φ_n(z) : n ∈ 𝒩}⊂(X∖X₀)∩φ(X∖X₀). Since φ(X∖X₀) is totally bounded, we can derive that {φ_n(z) : n ∈ 𝒩} converges to a point . Moreover, since φ is supercontractive. Then we have that and . By (44) we can see that . Since φ is supercontractive, there exists δ₁ > 0 such that

()

Choose N ∈ ℕ such that

for all n ≥ N, and we have, for any n ∈ ℕ, that

()

That is,

()

Since n is arbitrary, we can derive that f(φ_N(z)) = 0, and then f(z) = 0 since (44).

Case IV (ℱ ⊂ X∖X₀, ℱ is infinite and 𝒩 is finite). We can choose N₀ ∈ ℕ such that |h(φ_n(z))| < δ₀ for n > N₀. From (44), we have that

()

and then

()

for all n > N₀. This implies that f(z) = 0 as δ₀<|λ|.

Acknowledgments

The authors would like to express their thanks to the referees for several helpful comments which improved the presentation of this paper. Research of the second author was partially supported by NSF of China (11301285, 11371201). The fourth author was supported by Department of Applied Mathematics and the Research Group for Nonlinear Analysis and Optimization, post-doctoral fellowship at the National Sun Yat-sen University when this work was started, and this research is supported in part by Taiwan NSC grant 102-2115-M-033-006.

References

1 Garrido M. I. and Jaramillo J. A., Variations on the Banach-Stone theorem, Extracta Mathematicae. (2002) 17, no. 3, 351–383, ZBL1027.54021.
Google Scholar
2 Jarosz K. and Pathak V. D., K. Jarosz, Isometries and small bound isomorphisms of function spaces, Function Spaces, 1992, 136, Marcel Dekker, New York, NY, USA, 241–271, Lecture Notes in Pure and Applied Mathematics, ZBL0804.46030.
Google Scholar
3 Kamowitz H., Compact weighted endomorphisms of C(X), Proceedings of the American Mathematical Society. (1981) 83, no. 3, 517–521.
10.2307/2044109
Web of Science® Google Scholar
4 Cao J., Reilly I., and Xiong H., A lattice-valued Banach-Stone theorem, Acta Mathematica Hungarica. (2003) 98, no. 1-2, 103–110, 2-s2.0-0037256261, https://doi.org/10.1023/A:1022861506272, ZBL1027.46025.
10.1023/A:1022861506272
Web of Science® Google Scholar
5 Chen J.-X., Chen Z. L., and Wong N., A banach-stone theorem for riesz isomorphisms of banach lattices, Proceedings of the American Mathematical Society. (2008) 136, no. 11, 3869–3874, 2-s2.0-67650143563, https://doi.org/10.1090/S0002-9939-08-09582-8, ZBL1160.46026.
10.1090/S0002-9939-08-09582-8
Web of Science® Google Scholar
6 Ercan Z. and Önal S., Banach-stone theorem for Banach lattice valued continuous functions, Proceedings of the American Mathematical Society. (2007) 135, no. 9, 2827–2829, 2-s2.0-49549100353, https://doi.org/10.1090/S0002-9939-07-08788-6, ZBL1127.46026.
10.1090/S0002-9939-07-08788-6
Web of Science® Google Scholar
7 Ercan Z. and Önal S., The Banach-Stone theorem revisited, Topology and its Applications. (2008) 155, no. 16, 1800–1803, 2-s2.0-49549104185, https://doi.org/10.1016/j.topol.2008.05.018, ZBL1166.46309.
10.1016/j.topol.2008.05.018
Web of Science® Google Scholar
8 Miao X., Cao J., and Xiong H., Banach-Stone theorems and Riesz algebras, Journal of Mathematical Analysis and Applications. (2006) 313, no. 1, 177–183, 2-s2.0-28044464231, https://doi.org/10.1016/j.jmaa.2005.08.050, ZBL1102.46017.
10.1016/j.jmaa.2005.08.050
Web of Science® Google Scholar
9 Garrido M. I. and Jaramillo J. A., Homomorphisms on function lattices, Monatshefte fur Mathematik. (2004) 141, no. 2, 127–146, 2-s2.0-1542403817, https://doi.org/10.1007/s00605-002-0011-4, ZBL1058.54008.
10.1007/s00605-002-0011-4
Web of Science® Google Scholar
10 Garrido M. I. and Jaramillo J. A., Lipschitz-type functions on metric spaces, Journal of Mathematical Analysis and Applications. (2008) 340, no. 1, 282–290, 2-s2.0-37449024880, https://doi.org/10.1016/j.jmaa.2007.08.028, ZBL1139.46025.
10.1016/j.jmaa.2007.08.028
Web of Science® Google Scholar
11 Weaver N., Lattices of Lipschitz functions, Pacific Journal of Mathematics. (1994) 164, 179–193, https://doi.org/10.2140/pjm.1994.164.179, ZBL0797.46007.
10.2140/pjm.1994.164.179
Web of Science® Google Scholar
12 Jiménez-Vargas A. and Villegas-Vallecillos M., Order isomorphisms of little Lipschitz algebras, Houston Journal of Mathematics. (2008) 34, no. 4, 1185–1195, 2-s2.0-58549084584, ZBL1169.46010.
Web of Science® Google Scholar
13 Jiménez-Vargas A., Campoy A. M., and Villegas-Vallecillos M., The uniform separation property and banach-stone theorems for lattice-valued lipschitz functions, Proceedings of the American Mathematical Society. (2009) 137, no. 11, 3769–3777, 2-s2.0-77950799835, https://doi.org/10.1090/S0002-9939-09-09941-9, ZBL1202.46043.
10.1090/S0002-9939-09-09941-9
Web of Science® Google Scholar
14 Dubarbie L., Maps preserving common zeros between subspaces of vector-valued continuous functions, Positivity. (2010) 14, no. 4, 695–703, 2-s2.0-78649993953, https://doi.org/10.1007/s11117-010-0046-z, ZBL1220.47042.
10.1007/s11117-010-0046-z
Web of Science® Google Scholar
15 Araujo J. and Dubarbie L., Biseparating maps between Lipschitz function spaces, Journal of Mathematical Analysis and Applications. (2009) 357, no. 1, 191–200, 2-s2.0-67349244338, https://doi.org/10.1016/j.jmaa.2009.03.065, ZBL1169.47024.
10.1016/j.jmaa.2009.03.065
Web of Science® Google Scholar
16 Li L. and Leung D., Order isomorphisms on function space, Studia Mathematica. In press.
Google Scholar
17 Li L. and Wong N.-C., Kaplansky theorem of completely regular spaces, Proceedings of the American Mathematical Society. In press.
Google Scholar
18 Weaver N., Lipschitz Algebras, 1999, World Scientific Pbulishing, Singapore.
10.1142/4100
Google Scholar
19 Kamowitz H. and Scheinberg S., Some properties of endomorphisms of Lipschitz algebras, Studia Mathematica. (1990) 96, no. 3, 383–391, ZBL0713.47030.
10.4064/sm-96-3-255-261
Web of Science® Google Scholar
20 Jiménez-Vargas A. and Villegas-Vallecillos M., Compact composition operators on noncompact Lipschitz spaces, Journal of Mathematical Analysis and Applications. (2013) 398, no. 1, 221–229, https://doi.org/10.1016/j.jmaa.2012.08.044, ZBL06112308.
10.1016/j.jmaa.2012.08.044
Web of Science® Google Scholar

All articles

Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

Abstract

1. Introduction

2. Preliminaries

3. Nonvanishing Preservers on Lipschitz Functions

4. Compact Weighted Composition Operators on Lipschitz Spaces

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Nonvanishing Preservers and Compact Weighted Composition Operators between Spaces of Lipschitz Functions

Abstract

1. Introduction

2. Preliminaries

3. Nonvanishing Preservers on Lipschitz Functions

4. Compact Weighted Composition Operators on Lipschitz Spaces

Acknowledgments

References

References

Related

Information