Kamowitz, Herbert; Scheinberg, Stephen Some properties of endomorphisms of Lipschitz algebras. (English) Zbl 0713.47030 Stud. Math. 96, No. 3, 255-261 (1990). 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 \(\phi: K\to K\), then1) 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 Cited in 3 ReviewsCited in 18 Documents 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 Keywords:supercontraction PDFBibTeX XMLCite \textit{H. Kamowitz} and \textit{S. Scheinberg}, Stud. Math. 96, No. 3, 255--261 (1990; Zbl 0713.47030) Full Text: DOI EuDML