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

Abstract

By a theorem of Gordon and Hedenmalm, $\phi$ generates a bounded composition operator on the Hilbert space ${\mathcal{H}}^2$ of Dirichlet series ${\sum }_n{b}_n{n}^{-s}$ with square-summable coefficients $b_n$ if and 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 $\mathrm{Re}s>1/2$ 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 ${\mathcal{H}}^2$ 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 $\sup \mathrm{Re}\psi =1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\phi (s)=c_1+{\sum }_{j=1}^d{c}_{q_j}{q}_j^{-s}$ 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 ${\mathcal{H}}^2$ 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.

Final version, to appear in Journal of Functional Analysis

References in corpus (2)

• Decay rates for approximation numbers of composition operators
• On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

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• A mean counting function for Dirichlet series and compact composition operators
• Sharp norm estimates for composition operators and Hilbert-type inequalities
• Decay rates for approximation numbers of composition operators
• Norms of composition operators on the $H^2$ space of Dirichlet series
• Compact composition operators with non-linear symbols on the $H^2$ space of Dirichlet series
• Weak product spaces of Dirichlet series
• Composition operators on weighted Hilbert spaces of Dirichlet series
• Composition operators and embedding theorems for some function spaces of Dirichlet series
• Counting functions for Dirichlet series and compactness of composition operators
• An extension of Bohr's theorem
• Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series
• An embedding constant for the Hardy space of Dirichlet series
• Topological structure of the space of composition operators on the Hardy space of Dirichlet series
• Almost sure-sign convergence of Hardy-type Dirichlet series