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Some properties of endomorphisms of Lipschitz algebras. (English) Zbl 0713.47030
If (K,d) is a compact metric space, Lip(K,d) the Banach algebra of the functions f: $$K\to {\mathbb{C}}$$ for which $$\| f\|_{Lip(K,d)}=\| f\|_{\infty}+\sup_{x\neq y}[| f(x)-f(y)| /d(x,y)]<\infty$$ and T an endomorphism T: $$f\to f\circ \phi$$ of Lip(K,d) induced by a map $$\phi: K\to K$$, then
1) T is compact iff $$\phi$$ is a supercontraction, that is $$\lim_{d(x,y)\to 0}d(\phi (x),\phi (y))/d(x,y)=0$$ and
2) if $$T\neq 0$$ is compact then $$\sigma (T)=\{0,1\}$$, $$\sigma$$ (T) being the spectrum of T.
Reviewer: I.Gottlieb

##### MSC:
 47B38 Linear operators on function spaces (general) 47B07 Linear operators defined by compactness properties 54E45 Compact (locally compact) metric spaces 46J10 Banach algebras of continuous functions, function algebras
supercontraction
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