Composition operators on function spaces on the halfplane: spectra and semigroups

TL;DR

This paper studies the spectral properties and semigroup behavior of composition operators on Zen spaces, a class of weighted Bergman spaces on the right half-plane, extending previous work and illustrating results with Hardy--Bergman spaces.

Contribution

It generalizes existing results on norms, spectra, and semigroups of composition operators to Zen spaces, broadening understanding of their functional analysis properties.

Findings

Extended norm and essential norm estimates for composition operators.

Analyzed spectra and essential spectra of these operators.

Illustrated results with Hardy--Bergman space examples.

Abstract

This paper considers composition operators on Zen spaces (a class of weighted Bergman spaces of the right half-plane related to weighted function spaces on the positive half-line by means of the Laplace transform). Generalizations are given to work of Kucik on norms and essential norms, to work of Schroderus on (essential) spectra, and to work by Arvanitidis and the authors on semigroups of composition operators. The results are illustrated by consideration of the Hardy--Bergman space; that is, the intersection of the Hardy and Bergman Hilbert spaces on the half-plane.