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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Linear bijections preserving the Hölder seminorm
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by A. Jiménez-Vargas PDF
Proc. Amer. Math. Soc. 135 (2007), 2539-2547 Request permission

Abstract:

Let $(X,d)$ be a compact metric space and let $\alpha$ be a real number with $0<\alpha <1.$ The aim of this paper is to solve a linear preserver problem on the Banach algebra $C^{ {\alpha }}(X)$ of Hölder functions of order $\alpha$ from $X$ into $\mathbb {K}.$ We show that each linear bijection $T:C^{ {\alpha }} (X)\rightarrow C^{ {\alpha }}(X)$ having the property that $\alpha (T(f))=\alpha (f)$ for every $f\in C^{ {\alpha } }(X),$ where \begin{equation*} \alpha (f)=\sup \left \{ \frac {\left | f(x)-f(y)\right | }{d^{ {\alpha }} (x,y)}:x,y\in X,\ x\neq y\right \} , \end{equation*} is of the form $T(f)=\tau f\circ \varphi +\mu (f)1_X$ for every $f\in C^{ {\alpha } }(X),$ where $\tau \in \mathbb {K}$ with $\left | \tau \right | =1,$ $\varphi :X\rightarrow X$ is a surjective isometry and $\mu :C^{ {\alpha } }(X)\rightarrow \mathbb {K}$ is a linear functional.
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Additional Information
  • A. Jiménez-Vargas
  • Affiliation: Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04071, Almería, Spain
  • Email: ajimenez@ual.es
  • Received by editor(s): January 10, 2006
  • Received by editor(s) in revised form: February 13, 2006, and April 11, 2006
  • Published electronically: March 21, 2007
  • Additional Notes: This research was supported by Junta de Andalucia project P06-FQM-01438.
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2539-2547
  • MSC (2000): Primary 46E15; Secondary 46J10
  • DOI: https://doi.org/10.1090/S0002-9939-07-08756-4
  • MathSciNet review: 2302574