Eigenvalues and Stability Properties of Multiplication Operators and Multiplication Semigroups

TL;DR

This paper studies the stability properties of multiplication semigroups on Banach space valued L^p-spaces, linking their stability to pointwise semigroups and employing spectral mapping techniques.

Contribution

It provides a characterization of various stability types of multiplication semigroups via pointwise semigroups and proves a spectral mapping theorem for their spectra.

Findings

Stability properties can be characterized pointwise under certain conditions.

A spectral mapping theorem for point spectra is established.

Results facilitate stability analysis of multiplication semigroups on Banach spaces.

Abstract

We investigate uniform, strong, weak and almost weak stability of multiplication semigroups on Banach space valued $L^p$-spaces. We show that, under certain conditions, these properties can be characterized by analogous ones of the pointwise semigroups. Using techniques from selector theory, we prove a spectral mapping theorem for the point spectra of the pointwise and global semigroups and apply this as a major tool for determining almost weak stability.