Compact composition operators on noncompact Lipschitz spaces

Abstract

a b s t r a c tWe characterize compact composition operators between different spaces of scalar-valued Lipschitz functions defined on metric spaces, not necessarily compact, and determine their spectra.

Compact and weakly compact Lipschitz operators

“…Of course, by Schauder's theorem, f is compact if and only if f * is compact, so one can tackle the problem either working with C f or working with f . In [17], the authors proved the next characterization.…”

“…It is obvious that any compact operator is also weakly compact, while the converse is not true in general. A disguised study of compact Lipschitz operators has probably been initiated by Kamowitz and Scheinberg in [15] and then pursued by Jiménez-Vargas and Villegas-Vallecillos in [17] (see also [14] where vector-valued Lipschitz functions are considered). Indeed, in the last mentioned papers, the authors consider composition operators on Lipschitz spaces which appear naturally as the adjoints of our Lipschitz operators f .…”

“…Theorem ( [17,Theorem 1.2]). Let M be pointed separable metric spaces and let f : M → M be a Lipschitz map vanishing at 0 M .…”

Norm-attaining Composition Operators on Lipschitz Spaces

“…It is known (see [12,15]) that if X is a compact pointed metric space and ϕ : X → X is a basepoint-preserving Lipschitz map, then a composition operator C ϕ on lip 0 (X) is compact if and only if ϕ is supercontractive, that is,…”

“…Assuming that X is a compact metric space and ϕ is a Lipschitz map of X into X, Kamowitz and Scheinberg [15] proved that a composition operator C ϕ is compact on the spaces of bounded Lipschitz functions Lip(X) and lip(X α ) with the norm • ∞ + Lip(•) if and only if ϕ is supercontractive. This result was extended in [12] to composition operators on Lip 0 (X) when X is a bounded pointed metric space. Chen, Li, R. Wang and Y.-S. Wang [3] characterized compact weighted composition operators between spaces of scalar-valued Lipschitz functions.…”