Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series

TL;DR

This paper investigates the approximation numbers of composition operators on Hardy spaces of Dirichlet series, establishing bounds based on the form of the symbol and providing new estimates and criteria for compactness.

Contribution

It characterizes the approximation numbers of bounded composition operators on $H^p$ spaces of Dirichlet series and introduces new bounds, estimates, and a Littlewood--Paley formula for compactness.

Findings

Approximation numbers decay exponentially when $c_0=0$.

Polynomial decay of approximation numbers when $c_0>0$.

Explicit examples of compact operators with various decay rates.

Abstract

By a theorem of Bayart, $\varphi$ generates a bounded composition operator on the Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if $\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series with the following mapping properties: $\psi$ maps the right half-plane into the half-plane $\Real s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\Hp$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory ($s$-numbers, type and cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for $\Ht$, developed in an earlier paper, using estimates of solutions of the $\overline{\partial}$ equation. A transference principle from $H^p$ of the unit disc is discussed, leading to explicit examples of compact composition operators on $\Ho$ with approximation numbers decaying at a variety of sub-exponential rates. Finally, a new Littlewood--Paley formula is established, yielding a sufficient condition for a composition operator on $\Hp$ to be compact.