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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: $K\to {\bbfC}$ for which $\Vert f\Vert\sb{Lip(K,d)}=\Vert f\Vert\sb{\infty}+\sup\sb{x\ne y}[\vert f(x)-f(y)\vert /d(x,y)]<\infty$ 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 $\lim\sb{d(x,y)\to 0}d(\phi (x),\phi (y))/d(x,y)=0$ and 2) if $T\ne 0$ is compact then $\sigma (T)=\{0,1\}$, $\sigma$ (T) being the spectrum of T.
Reviewer: I.Gottlieb

47B38Operators on function spaces (general)
47B07Operators defined by compactness properties
54E45Compact (locally compact) metric spaces
46J10Banach algebras of continuous functions, function algebras
Full Text: EuDML