Unified approach to spectral properties of multipliers

Abstract

Let ${\mathbb{B}}_n$ be the open unit ball in ${\mathbb{C}}^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on ${\mathbb{B}}_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A_{\alpha ,\beta }^p$, Bloch-type spaces ${\mathcal{B}}_{\alpha }$, weighted Hardy spaces $H_w^p$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H_{\beta }^2$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.

19 pages. To appear in Taiwanese J. Math