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 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 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 .
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