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Proceedings of the American Mathematical Society
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
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

$\displaystyle \alpha (f)=\sup \left\{ \frac{\left\vert f(x)-f(y)\right\vert }{d^{ {\alpha}} (x,y)}:x,y\in X, x\neq y\right\} , $

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\vert \tau \right\vert =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

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08756-4
PII: S 0002-9939(07)08756-4
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.