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

Let (X,d) be a metric space and Lip (X,d) denote the space of all bounded Lipschitz functions f on the metric space (X,d). A mapping ϕ:XX is called supercontractive if, for ϵ>0, there exists δ>0 such that d(ϕ(x),ϕ(y))<ϵd(x,y) whenever 0<d(x,y)<δ. The authors prove that C ϕ , the composition operator defined by C ϕ f=fϕ, is compact on Lip (X,d) if and only if ϕ is supercontractive and ϕ(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:
47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions