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

Abstract

By a theorem of the first named author, $\varphi $ generates a bounded composition operator on the Hardy space ${\mathscr{H}}^p$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 $\operatorname{Re} s >1/2$ 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 ${\mathscr{H}}^p$ are bounded below by a constant times $r^n$ for some $00$ 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 ${\mathscr{H}}^2$, 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 ${\mathscr{H}}^1$ 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 ${\mathscr{H}}^p$ to be compact.