• Hölder Seminorm Preserving Linear Bijections and Isometries

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/01/0053-0062

• # Keywords

Lipschitz function; isometry; linear preserver problem; Banach-Stone theorem.

• # Abstract

Let $(X, d)$ be a compact metric and $0 &lt; \alpha &lt; 1$. The space $\mathrm{Lip}^\alpha(X)$ of Hölder functions of order 𝛼 is the Banach space of all functions 𝑓 from 𝑋 into $\mathbb{K}$ such that $\| f\|=\max \{\| f\|_\infty,L(f)\} &lt;\infty$, where

$$L(f)=\sup\{|f(x)-f(y)|/d^\alpha(x,y):x,y\in X, x\neq y\}$$

is the Hölder seminorm of 𝑓. The closed subspace of functions 𝑓 such that

$$\lim\limits_{d(x,y)\to 0}|f(x)-f(y)|/d^\alpha(x,y)=0$$

is denoted by $\mathrm{lip}^\alpha(X)$. We determine the form of all bijective linear maps from $\mathrm{lip}^\alpha(X)$ onto $\mathrm{lip}^\alpha(Y)$ that preserve the Hölder seminorm.

• # Author Affiliations

1. Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120, Almería, Spain

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

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