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 : H H(Ω) the associated composition operator defined by
Cϕ(f) = f ϕ.
We want to know when 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 :
2010 Mathematics Subject Classification. Primary 47B33, 30B50, 30H10.
c 2015 American Mathematical Society
1
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