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

By a theorem of Bayart, $\phi$ generates a bounded composition operator on the Hardy space $\text{Hp}$of Dirichlet series ($1\le p<\mathrm{\infty }$) only if $\phi (s)=c_{0}s+\psi (s)$, 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 $\text{Real}s>1\mathrm{/}2$ 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 $\text{Hp}$ 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 $\text{Ht}$, developed in an earlier paper, using estimates of solutions of the $\bar{\mathrm{\partial }}$ equation. A transference principle from $H^p$ of the unit disc is discussed, leading to explicit examples of compact composition operators on $\text{Ho}$ 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 $\text{Hp}$ to be compact.