Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46E15, 46J10

Retrieve articles in all journals with MSC (2000): 46E15, 46J10

Additional Information

A. Jiménez-Vargas
Affiliation: Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04071, Almería, Spain

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.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia