Compact endomorphisms of certain analytic Lipschitz algebras

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.