2 H.
QUEFFELEC´
H H, in particular its spectrum, compactness, membership in a Schatten class,
and more precisely the decay rate of its approximation numbers to be defined. One
basic example is that of the Hardy space
H2(D)
= f(z) =

n=0
bnzn
: f
2
:=

n=0
|bn|2
,
a Hilbert space of analytic functions on the open unit disk D, with orthonormal
basis en(z) =
zn,
n = 0,... and hence reproducing kernel Ka(z) =
1
1−az
. In that
case, all analytic self-maps ϕ : D D are symbols ([31]).
By definition, the subject is border-line between soft analysis (here operator
theory) and hard analysis (Carleson measures, interpolation sequences, d-bar cor-
rection). Moreover, some proofs are rather delicate and long. And accordingly this
survey will only detail the involved tools and one proof. It is roughly divided into
five general parts:
(1) The specific Hilbert space involved and its composition operators
(2) Definitions and tools from operator theory (Weyl’s inequalities, . . . )
(3) Definitions and tools from function theory (Carleson measures,. . . )
(4) General estimates for approximation numbers
(5) Statement of the main results and proof of one.
2. The space H
2
of Dirichlet series
We will denote by the half-plane = {s : Re s θ}, θ R. The Hilbert
space H
2
consists of all ordinary Dirichlet series f(s) =


n=1
bnn−s
such that
f
2
H 2
:=

n=1
|bn|2
∞.
This is clearly a Hilbert space of analytic functions on C1/2 (by the Cauchy-Schwarz
inequality) with orthonormal basis en, en(s) =
n−s,
n = 1, 2,... and therefore
reproducing kernel
Ka(s) =

n=1
en(s)en(a) = ζ(s + a)
where ζ is the Riemann zeta function.
This space was (re)-introduced in 1997 by Hedenmalm, Lindqvist, Seip ([13]) to
characterize those functions f(t) =
∑∞
n=1

2an sin 2πnt whose dilates fk(t) = f(kt)
form a Riesz sequence (an essential issue in the present work) in
L2(0,
1). The
authors showed (see also [6]) that the space of multipliers of H 2 is (isometrically)
the space H∞ of Dirichlet series which are bounded in C0, and hence convergent in
this half-plane thanks to a result of Bohr ([4]). Their main theorem is as follows
Theorem 2.1 ([13]). Let G(s) =
∑∞
n=1
ann−s be the Dirichlet generating func-
tion of f(t) =
∑∞
n=1

2an sin 2πnt. Then, the following are equivalent:
1. (fk) is a Riesz sequence in
L2(0,
1).
2. G and 1/G belong to
H∞.
We now describe the admissible composition operators on H
2.
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