Weighted holomorphic Dirichlet series and composition operators with polynomial symbols

TL;DR

This paper introduces weighted spaces of holomorphic Dirichlet series with real frequencies, analyzing composition operators with polynomial symbols, and provides criteria for their boundedness, compactness, and cyclicity properties.

Contribution

It develops a new framework for weighted holomorphic Dirichlet spaces and characterizes composition operators with polynomial symbols, including affine functions.

Findings

Criteria for boundedness and compactness of composition operators with affine symbols.

Conditions for cyclicity of certain operators on Hardy spaces.

Extension of the theory of composition operators to weighted Dirichlet series spaces.

Abstract

In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H^2 is cyclic.