Weighted composition operators on spaces of analytic vector-valued Lipschitz functions

TL;DR

This paper characterizes when weighted composition operators are bounded or compact on spaces of vector-valued Lipschitz functions, extending operator theory in complex analysis.

Contribution

It provides necessary and sufficient conditions for boundedness and compactness of weighted composition operators on vector-valued Lipschitz spaces.

Findings

Characterization of boundedness conditions

Criteria for compactness of operators

Extension to vector-valued Lipschitz function spaces

Abstract

Let {\phi} be an analytic self-map of D and be an analytic operator-valued function on D, where D is the unit disk. We provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators W_{\psi,\phi} on Lip_A(D;X;\alpha) and lip_A(D;X;{\alpha}), the spaces of analytic X-valued Lipschitz functions f, where X is a complex Banach space and {\alpha} is in (0, 1].