On composition operators on the Wiener algebra of Dirichlet series

TL;DR

This paper demonstrates that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series can lie outside the algebra, providing an example of an absolutely summing and compact operator.

Contribution

It reveals that the symbol of a bounded composition operator need not be in the Wiener algebra, with an explicit example of an absolutely summing, compact operator.

Findings

The symbol of a bounded composition operator can be outside the Wiener algebra.

An explicit example of an absolutely summing, compact composition operator is provided.

The results challenge previous assumptions about symbols of such operators.

Abstract

We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.