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Compact composition operators on noncompact Lipschitz spaces. (English) Zbl 1279.47042

Let $\left(X,d\right)$ be a metric space and $\mathrm{Lip}\left(X,d\right)$ denote the space of all bounded Lipschitz functions $f$ on the metric space $\left(X,d\right)$. A mapping $\varphi :X\to X$ is called supercontractive if, for $ϵ>0$, there exists $\delta >0$ such that $d\left(\varphi \left(x\right),\varphi \left(y\right)\right)<ϵd\left(x,y\right)$ whenever $0. The authors prove that ${C}_{\varphi }$, the composition operator defined by ${C}_{\varphi }f=f\circ \varphi$, is compact on $\mathrm{Lip}\left(X,d\right)$ if and only if $\varphi$ is supercontractive and $\varphi \left(X\right)$ is totally bounded in $X$.

This result improves upon previous work of H. Kamowitz and S. Scheinberg [Stud. Math. 96, No. 3, 255–261 (1990; Zbl 0713.47030)] that studied the case of compact metric spaces.

##### MSC:
 47B33 Composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions