Approximation numbers of composition operators on the $H^2$ space of Dirichlet series

TL;DR

This paper investigates the approximation numbers of bounded composition operators on the Hilbert space of Dirichlet series, revealing their decay rates and providing explicit examples with various compactness properties.

Contribution

It characterizes the decay rates of approximation numbers for composition operators on $\

Findings

Approximation numbers decay exponentially when $c_0=0$ and polynomially when $c_0>0$.

Explicit examples of compact operators with sub-exponential decay are constructed.

The paper introduces a Hilbert space method and an interpolation approach using the $ar{ ext{d}}$-equation for analysis.

Abstract

By a theorem of Gordon and Hedenmalm, $\varphi$ generates a bounded composition operator on the Hilbert space $\mathscr{H}^2$ of Dirichlet series $\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and 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 $\operatorname{Re} s >1/2$ 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 $\mathscr{H}^2$ 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. The case when $c_0=0$, $\psi$ is bounded and smooth up to the boundary of the right half-plane, and $\sup \operatorname{Re} \psi=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\varphi(s)=c_1+\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\mathscr{H}^2$ using estimates of solutions of the $\bar{\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.