APPROXIMATION NUMBERS OF COMPOSITION OPERATORS 3

3. Bounded composition operators on H

2

The Gordon–Hedenmalm theorem ([12]) gives a full description of those an-

alytic maps ϕ : C1/2 → C1/2 which generate a composition operator on H

2

and

reads as follows, under a slightly reinforced form ([25]):

Theorem 3.1 ([12]). The function ϕ determines a bounded composition oper-

ator Cϕ on H 2 if and only if

ϕ(s) = c0s +

∞

n=1

cnn−s

=: c0s + ψ(s),

where c0 is a nonnegative integer and ψ is a Dirichlet series that converges uni-

formly in Cε for every ε 0 and has the following mapping properties:

(a) If c0 = 0, then ψ(C0) ⊂ C1/2.

(b) If c0 ≥ 1, then either ψ ≡ 0 or ψ(C0) ⊂ C0.

We thus see that there are few admissible symbols ϕ and that, depending on the

value of a weird parameter c0, they have more or less restrictive mapping properties

(observe that c0 ≥ 1 corresponds to ϕ(∞) = ∞). Also note that this reformulation

of the initial result is slightly stronger, in that we claim the uniform convergence

of ψ in each half-plane Cε, not only in some half-plane Cσ0 , and that the mapping

properties are stated in terms of ψ rather than in terms of ϕ, which is stronger and

was also proved, in a different way, in the paper [12].

4. Definitions and tools from operator theory

4.1. Approximation numbers. Let T : H → H and n ≥ 1. The n-th

approximation number of T is: an(T ) = inf{T − R ; rank R n}. Then (short

list, see [5, p. 155]):

1. T = a1(T ) ≥ a2(T ) ≥ ··· ≥ an(T ) ≥ ···

2. T is compact iff an(T ) ↓ 0. Otherwise (take T = Diag εn), the sequence

(an(T )) can be any arbitrary non-increasing sequence.

3. an(T ) = an(T ∗).

4. an(ATB) ≤ A an(T ) B (Ideal property).

5. an(T ) = sn(T ) (singular number, Allakhverdiev, 1957).

6. Sp = {T ∈ L(H) ;

∑∞

n=1

(an(T

))p

1/p

∞} (Schatten class Sp, p 0).

Note that p q ⇒ Sp ⊂ Sq.

The decay rate of an(T ) (non-commutative approximation theory) measures

the degree of compactness of T .

We recall that (Schmidt decomposition, see [5], p.46) a compact operator T :

H → H can be written under the form T (x) =

∑∞

n=1

sn(T ) x, vn un where (un)

and (vn) are two orthonormal sequences, and that we have the equations

(1) T (vn) = sn(T )un, sn(T ) = an(T ) = T (vn),un .