Bulletin of the Belgian Mathematical Society - Simon Stevin

Compact endomorphisms of certain analytic Lipschitz algebras

F. Behrouzi and H. Mahyar

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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.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 2 (2005), 301-312.

Dates
First available in Project Euclid: 3 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1117805091

Digital Object Identifier
doi:10.36045/bbms/1117805091

Mathematical Reviews number (MathSciNet)
MR2179971

Zentralblatt MATH identifier
1110.46036

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]

Keywords
compact endomorphisms Lipschitz algebras analytic functions spectra

Citation

Behrouzi, F.; Mahyar, H. Compact endomorphisms of certain analytic Lipschitz algebras. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 2, 301--312. doi:10.36045/bbms/1117805091. https://projecteuclid.org/euclid.bbms/1117805091


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