Approximation numbers of composition operators on the [formula omitted] space of Dirichlet series

Abstract

By a theorem of Gordon and Hedenmalm, φ generates a bounded composition operator on the Hilbert space H2 of Dirichlet series ∑nbnn−s with square-summable coefficients bn if and only if φ(s)=c0s+ψ(s), where c0 is a nonnegative integer and ψ a Dirichlet series with the following mapping properties: ψ maps the right half-plane into the half-plane Res>1/2 if c0=0 and is either identically zero or maps the right half-plane into itself if c0 is positive. It is shown that the nth approximation numbers of bounded composition operators on H2 are bounded below by a constant times rn for some 0<r<1 when c0=0 and bounded below by a constant times n−A for some A>0 when c0 is positive. Both results are best possible. The case when c0=0, ψ is bounded and smooth up to the boundary of the right half-plane, and sup⁡Reψ=1/2, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes Sp. For φ(s)=c1+∑j=1dcqjqj−s with qj independent integers, it is shown that the nth 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 H2 using estimates of solutions of the ∂¯ equation. Finally, by a transference principle from H2 of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.