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

Dongyang Chen,Risheng Wang,Ya-Shu Wang,Lei Li

doi:10.1155/2013/741050

Abstract

We will give the -Lipschitz version of the Banach-Stone type theorems for lattice-valued -Lipschitz functions on some metric spaces. In particular, when and are bounded metric spaces, if is a nonvanishing preserver, then is a weighted composition operator , where is a Lipschitz homeomorphism. We also characterize the compact weighted composition operators between spaces of Lipschitz functions.

Highlights

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 (Tf) (y) = J (y) f (φ (y)), (1)
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
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

Summary

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 (Tf) (y) = J (y) f (φ (y)) ,. 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. When X, Y are bounded metric spaces, if T : Lip(X) → Lip(Y) is a nonvanishing preserver, 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

Preliminaries

Nonvanishing Preservers on Lipschitz Functions

Compact Weighted Composition Operators on Lipschitz Spaces