Multiplication operators on the Bloch space of bounded homogeneous domains

TL;DR

This paper investigates multiplication operators on the Bloch space of bounded homogeneous domains, characterizing boundedness, compactness, spectrum, and isometric operators, with specific results for polydisks and symmetric domains.

Contribution

It provides a complete characterization of bounded, compact, and isometric multiplication operators on the Bloch space of bounded homogeneous domains, including polydisks and symmetric domains.

Findings

Only constant symbols induce bounded multiplication operators on the polydisk.

Isometric multiplication operators have symbols of constant modulus one on certain domains.

The spectrum of multiplication operators is explicitly determined.

Abstract

In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. We prove that the only bounded multiplication operators on the Bloch space of the polydisk are those whose symbol is constant. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.