Compact composition operators with nonlinear symbols on the $H^2$ space of Dirichlet series

Frédéric Bayart and Ole Fredrik Brevig

Vol. 291 (2017), No. 1, 81–120

DOI: 10.2140/pjm.2017.291.81

Abstract

We investigate compactness of composition operators on the Hardy space of Dirichlet series induced by a map $\phi \left(s\right)=c_{0}s+{\phi }_0\left(s\right)$, where ${\phi }_0$ is a Dirichlet polynomial. Our results depend heavily on the characteristic $c_0$ of $\phi$ and, when $c_0=0$, on both the degree of ${\phi }_0$ and its local behavior 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.

Keywords

Dirichlet series, composition operators, approximation numbers

Mathematical Subject Classification 2010

Primary: 47B33

Secondary: 30B50, 30H10

Milestones

Received: 17 November 2015

Accepted: 22 November 2016

Published: 20 August 2017