Compact composition operators with non-linear symbols on the $H^2$ space of Dirichlet series

TL;DR

This paper studies the compactness of certain composition operators on the Hardy space of Dirichlet series, focusing on the influence of the symbol's linear part and local boundary behavior, using geometric and differential tools.

Contribution

It characterizes the compactness of these operators based on the symbol's linear coefficient and polynomial degree, introducing new geometric and differential geometric techniques.

Findings

Compactness depends on the characteristic $c_0$ and boundary behavior.

Approximation numbers are analyzed for specific operators.

Geometric estimates of Carleson measures are employed.

Abstract

We investigate the compactness of composition operators on the Hardy space of Dirichlet series induced by a map $\varphi(s)=c_0s+\varphi_0(s)$, where $\varphi_0$ is a Dirichlet polynomial. Our results depend heavily on the characteristic $c_0$ of $\varphi$ and, when $c_0=0$, on both the degree of $\varphi_0$ and its local behaviour near a boundary point. We also study the approximation numbers for some of these operators. Our methods involve geometric estimates of Carleson measures and tools from differential geometry.