Bulletin of the Belgian Mathematical Society - Simon Stevin

Compact endomorphisms of certain analytic Lipschitz algebras

F. Behrouzi and H. Mahyar

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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

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

First available in Project Euclid: 3 June 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

compact endomorphisms Lipschitz algebras analytic functions spectra


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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