Translator Disclaimer
June 2005 Compact endomorphisms of certain analytic Lipschitz algebras
F. Behrouzi, H. Mahyar
Bull. Belg. Math. Soc. Simon Stevin 12(2): 301-312 (June 2005). DOI: 10.36045/bbms/1117805091

Abstract

Let $X$ be a compact plane set. $A(X)$ denotes the uniform algebra of all continuous complex-valued functions on $X$ which are analytic on int$X$. For $0<\alpha\leq 1$, Lipschitz algebra of order $\alpha$, $Lip(X,\alpha)$ is the algebra of all complex-valued functions $f$ on $X$ for which $p_\alpha(f)=\sup\{\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}:x,y\in X, x\neq y\}<\infty.$ Let $Lip_A(X,\alpha)=A(X)\bigcap Lip(X,\alpha)$, and $Lip^{n}(X,\alpha)$ be the algebra of complex-valued functions on $X$ whose derivatives up to order $n$ are in $\Lip(X,\alpha)$. $Lip_A(X,\alpha)$ under the norm $\|f\|=\|f\|_X+p_\alpha(f)$, and $Lip^n(X,\alpha)$ for a certain plane set $X$ under the norm $\|f\|=\sum_{k=0}^{n}\frac{\|f^{(k)}\|_X+p_{\alpha}(f^{(k)})}{k!}$ are natural Banach function algebras, where $\|f\|_X = \sup_{x\in X } |f(x)|$. In this note we study endomorphisms of algebras $Lip_A(X,\alpha)$ and $Lip^n(X,\alpha)$ and investigate necessary and sufficient conditions for which these endomorphisms to be compact. Finally, we determine the spectra of compact endomorphisms of these algebras.

Citation

Download Citation

F. Behrouzi. H. Mahyar. "Compact endomorphisms of certain analytic Lipschitz algebras." Bull. Belg. Math. Soc. Simon Stevin 12 (2) 301 - 312, June 2005. https://doi.org/10.36045/bbms/1117805091

Information

Published: June 2005
First available in Project Euclid: 3 June 2005

zbMATH: 1110.46036
MathSciNet: MR2179971
Digital Object Identifier: 10.36045/bbms/1117805091

Subjects:
Primary: 46J10
Secondary: ‎46J15

Rights: Copyright © 2005 The Belgian Mathematical Society

JOURNAL ARTICLE
12 PAGES


SHARE
Vol.12 • No. 2 • June 2005
Back to Top