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 Cθ the half-plane Cθ = {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.