Composition operators in the Lipschitz Space of the Polydiscs

TL;DR

This paper generalizes Shapiro's 1987 result on the compactness of composition operators induced by holomorphic self-maps to the Lipschitz space of polydiscs in higher dimensions, using spectral theory.

Contribution

It extends the characterization of compact composition operators from the unit disk to the polydisc in multiple dimensions.

Findings

Generalization of Shapiro's result to n-dimensional polydiscs

Spectral-theoretic characterization of compactness in higher dimensions

Conditions on the symbol function for compactness

Abstract

In 1987, Shapiro shew that composition operator induced by symbol $\phi$ is compact on the Lipschltz space if and only if the infinity norm of $\phi$ is less than 1 by a spectral-theoretic argument, where $\phi$ is a holomorphic self-map of the unit disk. In this paper, we shall generalize Shapiro's result to the $n$-dimensional case.