Contemporary Mathematics

Volume 638, 2015

http://dx.doi.org/10.1090/conm/638/12802

Approximation numbers of composition operators

on a Hilbert space of Dirichlet series

Herv´ e Queff´elec

Abstract. By a theorem of Gordon and Hedenmalm, ϕ generates a bounded

composition operator on the Hilbert space H

2

of Dirichlet series

∑

n

bnn−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 number of a bounded com-

position operator on H 2 is bounded below by a constant times rn for some

0 r 1 when c0 = 0 and by a constant times n−A for some A 0 when c0 is

positive. Both results are best possible. The case ϕ(s) = c1 +

∑d

j=1

cqj qj

−s

is

mentioned. In that case, the nth approximation number behaves as

n−(d−1)/2,

possibly up to a factor (log

n)(d−1)/2.

Finally, a general transference princi-

ple and recent results in the usual Hardy space allow us to exhibit compact

composition operators on H

2

whose approximation numbers decay arbitrarily

slowly. Estimates rely mainly on a general Hilbert space method involving re-

producing kernels. A key role is played by a recently developed interpolation

method for H

2 using estimates of solutions of the ∂ equation.

1. Introduction and statement of main results

This survey is mainly based on the paper [25]. Let Ω be an open subset of C,

H(Ω) the space of analytic functions on Ω with its natural topology, and H ⊂ H(Ω)

be a separable Hilbert space with a specified orthonormal basis (en). We assume

that H is continuously embedded in H(Ω), which amounts to say that the point

evaluations δa, a ∈ Ω are continuous on H and therefore given by a scalar product:

δa(f) = f(a) = f, Ka , where Ka ∈ H.

The function Ka is called the reproducing kernel of H at a and classically given by

Ka(z) = en(z)en(a)

with convergence of the series in H. Let now ϕ : Ω → Ω be an analytic self-map

and Cϕ : H → H(Ω) the associated composition operator defined by

Cϕ(f) = f ◦ ϕ.

We want to know when Cϕ actually maps H to itself (then, ϕ is called a “symbol”),

and to compare the properties of the function ϕ : Ω → Ω and the operator Cϕ :

2010 Mathematics Subject Classification. Primary 47B33, 30B50, 30H10.

c 2015 American Mathematical Society

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