Topological structure of the space of composition operators on the Hardy space of Dirichlet series

TL;DR

This paper investigates the topological relationships between composition operators on the Hardy space of Dirichlet series, focusing on their components and compactness properties, with implications for operators with polynomial symbols.

Contribution

It characterizes when two such operators are in the same component and establishes that compactness of polynomial combinations implies individual compactness.

Findings

Two composition operators are in the same component if their difference is not compact.

A linear combination of polynomial-symbol composition operators is compact only if each operator is compact.

The results connect the topological structure of the operator space with compactness criteria.

Abstract

The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.