Isometries and spectra of multiplication operators on the Bloch space

TL;DR

This paper characterizes isometric multiplication operators on the Bloch space, describes their spectra in terms of the symbol's range, and analyzes specific weighted composition operators, providing bounds and classifications.

Contribution

It provides a complete characterization of isometric multiplication operators and their spectra on the Bloch space, including a detailed analysis of weighted composition operators.

Findings

Isometric multiplication operators are induced by constant functions of modulus 1.

The spectrum of a multiplication operator is determined by the range of its symbol.

Identifies isometries and spectra of certain weighted composition operators.

Abstract

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.