Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

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→0 sup⁡0<d(x, y)&...

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Main Authors: A. Jiménez-Vargas, Miguel Lacruz, Moisés Villegas-Vallecillos
Format: Article
Language:English
Published: Hindawi Limited 2011-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2011/590853
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spelling doaj-ed2043ed99d842f7bd86ea78d59d289b2020-11-24T20:46:27ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/590853590853Essential Norm of Composition Operators on Banach Spaces of Hölder FunctionsA. Jiménez-Vargas0Miguel Lacruz1Moisés Villegas-Vallecillos2Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, SpainDepartamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina, Mercedes s/n, 41012 Sevilla, SpainDepartamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, SpainLet (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→0 sup⁡0<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.http://dx.doi.org/10.1155/2011/590853
collection DOAJ
language English
format Article
sources DOAJ
author A. Jiménez-Vargas
Miguel Lacruz
Moisés Villegas-Vallecillos
spellingShingle A. Jiménez-Vargas
Miguel Lacruz
Moisés Villegas-Vallecillos
Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
Abstract and Applied Analysis
author_facet A. Jiménez-Vargas
Miguel Lacruz
Moisés Villegas-Vallecillos
author_sort A. Jiménez-Vargas
title Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
title_short Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
title_full Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
title_fullStr Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
title_full_unstemmed Essential Norm of Composition Operators on Banach Spaces of Hölder Functions
title_sort essential norm of composition operators on banach spaces of hölder functions
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2011-01-01
description 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→0 sup⁡0<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.
url http://dx.doi.org/10.1155/2011/590853
work_keys_str_mv AT ajimenezvargas essentialnormofcompositionoperatorsonbanachspacesofholderfunctions
AT miguellacruz essentialnormofcompositionoperatorsonbanachspacesofholderfunctions
AT moisesvillegasvallecillos essentialnormofcompositionoperatorsonbanachspacesofholderfunctions
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