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Proceedings of the American Mathematical Society

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

 
 

 

Linear bijections preserving the Hölder seminorm


Author: A. Jiménez-Vargas
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
Published electronically: March 21, 2007
MathSciNet review: 2302574
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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

Keywords: Linear preserver problem, extreme point, isometry.
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.