Estimate for norm of a composition operator on the Hardy-Dirichlet space

TL;DR

This paper provides bounds on the norm of a specific composition operator on the Hardy-Dirichlet space using the Schur test, and estimates its approximation numbers, advancing understanding of operator behavior in this function space.

Contribution

It introduces new upper and lower bounds for the norm of composition operators with particular symbols on the Hardy-Dirichlet space, and estimates their approximation numbers.

Findings

Derived bounds for the operator norm using the Schur test.

Provided estimates for the approximation numbers of the operator.

Focused on symbols of the form c_1 + c_q q^{-s} with fixed q.

Abstract

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $\mathcal{H}^2$, the space of Dirichlet series with square summable coefficients, for the inducing symbol $\varphi(s)=c_1+c_{q}q^{-s}$ where $q\geq 2$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.