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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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[1] Kamowitz, H.; Scheinberg, S., Some properties of endomorphisms of Lipschitz algebras, Studia Math., 96, 61-67, (1990) · Zbl 0713.47030
[2] Araujo, J.; Dubarbie, L., Biseparating maps between Lipschitz function spaces, J. Math. Anal. Appl., 357, 191-200, (2009) · Zbl 1169.47024
[3] Weaver, N., Isometries of noncompact Lipschitz spaces, Canad. Math. Bull., 38, 242-249, (1995) · Zbl 0831.46007
[4] Araujo, J.; Dubarbie, L., Noncompactness and noncompleteness in isometries of Lipschitz spaces, J. Math. Anal. Appl., 377, 15-29, (2011) · Zbl 1231.47030
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