Abstract and Applied Analysis
Volume 2011 (2011), Article ID 590853, 13 pages
http://dx.doi.org/10.1155/2011/590853
Research Article

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

A. Jiménez-Vargas,1 Miguel Lacruz,2 and Moisés Villegas-Vallecillos1

1Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Avenida Reina, Mercedes s/n, 41012 Sevilla, Spain

Received 29 June 2011; Revised 30 September 2011; Accepted 30 September 2011

Academic Editor: Simeon Reich

Copyright © 2011 A. Jiménez-Vargas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 l i p 0 ( 𝑋 , 𝑑 𝛼 ) and L i p 0 ( 𝑋 , 𝑑 𝛼 ) is given by the formula ‖ 𝐶 𝜑 ‖ 𝑒 = l i m 𝑡 → 0 s u p 0 < 𝑑 ( 𝑥 , 𝑦 ) < 𝑡 ( 𝑑 ( 𝜑 ( 𝑥 ) , 𝜑 ( 𝑦 ) ) 𝛼 / 𝑑 ( 𝑥 , 𝑦 ) 𝛼 ) whenever the dual space l i p 0 ( 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.