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Compact composition operators on noncompact Lipschitz spaces. (English) Zbl 1279.47042
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 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 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions
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##### References:
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