Compact composition operators on noncompact Lipschitz spaces (Q691827)

Let \((X,d)\) be a metric space and \(\mathrm{Lip}(X,d)\) denote the space of all bounded Lipschitz functions \(f\) on the metric space \((X,d)\). A mapping \(\phi:X\to X\) is called supercontractive if, for \(\epsilon>0\), there exists \(\delta>0\) such that \(d(\phi(x),\phi(y))<\epsilon d(x,y)\) whenever \(0<d(x,y)<\delta\). The authors prove that \(C_\phi\), the composition operator defined by \(C_\phi f=f\circ \phi\), is compact on \(\mathrm{Lip}(X,d)\) if and only if \(\phi\) is supercontractive and \(\phi(X)\) is totally bounded in \(X\). This result improves upon previous work of \textit{H. Kamowitz} and \textit{S. Scheinberg} [Stud. Math. 96, No.~3, 255--261 (1990; Zbl 0713.47030)] that studied the case of compact metric spaces.