This is an announcement for the paper "Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series" by Herve Queffelec, and Kristian Seip.
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 ecotype 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.
Archive classification: math.FA math.CV
Submitted from: seip@math.ntnu.no
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1406.0445
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