Composition operators in the Lipschitz Space of the Polydiscs

Abstract

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