Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators

TL;DR

This paper proves that multipliers of certain function spaces cannot reach their maximum norm unless constant, and explores properties of multiplication operators such as being constant, non-unitary, compact, or mean ergodic.

Contribution

It establishes a maximum modulus principle for multipliers on normed spaces of continuous functions and analyzes conditions for compactness and mean ergodicity of multiplication operators.

Findings

Multipliers cannot attain their norms unless they are constant.

A contractive multiplication operator is either constant or completely non-unitary.

Conditions for multiplication operators to be compact and mean ergodic are characterized.

Abstract

The main goal of this note is to show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants. As an application, a contractive multiplication operator is either a multiplication with a constant, or is completely non-unitary. Additionally we explore possibilities for a multiplication operator to be (weakly) compact and (uniformly) mean ergodic.