1. Home
    2. Journals
    3. Proceedings – Mathematical Sciences
    4. Volume 119
    5. Issue 1

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      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 $0 < \alpha < 1$. 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 $\| f\|=\max \{\| f\|_\infty,L(f)\} <\infty$, 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

       

      A Jiménez-Vargas1 M A Navarro1

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

    • Dates

       
      Published
      February 2009
      Manuscript received
      21 March 2007
      Manuscript revised
      26 July 2008

  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2022-2023 Indian Academy of Sciences, Bengaluru.

Contact | Site index