Composition operators on weighted Hilbert spaces of Dirichlet series

TL;DR

This paper investigates the properties of composition operators on weighted Hilbert spaces of Dirichlet series, establishing conditions for their boundedness and compactness through new analytical tools and a Schwarz-type lemma.

Contribution

It introduces weighted mean counting functions and a change of variables formula, providing new criteria for boundedness and compactness of composition operators on these spaces.

Findings

Necessary conditions for boundedness and compactness are established.

For Bergman-type spaces, compactness conditions are also sufficient.

A Schwarz-type lemma for Dirichlet series is developed.

Abstract

We study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman-type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz-type lemma for Dirichlet series.