Composition operators and generalized primes

TL;DR

This paper investigates the compactness of composition operators on Hardy spaces of Dirichlet series, introducing a generalized setting based on Beurling's primes and establishing a necessary condition using a Nevanlinna-type counting function.

Contribution

It extends the analysis of composition operators to a generalized Hardy space framework involving Beurling's primes, providing new necessary conditions for compactness.

Findings

Derived a Nevanlinna-type necessary condition for compactness

Extended Hardy space concepts to generalized Dirichlet series

Linked composition operator properties to prime sequence structures

Abstract

We study composition operators on the Hardy space $\mathcal{H}^2$ of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on $\mathcal{H}^2$. To do that we extend our notions to a Hardy space $\mathcal{H}_{\Lambda}^2$ of generalized Dirichlet series, induced in a natural way by a sequence of Beurling's primes.