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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
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