Page 33 - Textos de Matemática Vol. 34

INTERPOLATION THEORY AND S-NUMBERS 23
bounded operator from Ei into Fi, i = 0, 1. The space L (E, F ) is a Banach space with the norm
(2.1) ∥T∥E,F = max{∥T∥E0,F0,∥T∥E1,F1}.
When E0 = E1 = E, then we write L(E,F) instead of L(E,F). Similarly, we write L(E,F) instead of L(E,F) when F0 = F1 = F.
If E is an intermediate space with respect to E = (E0, E1), then for each t > 0 we set
and
ψ(t) = ψ(t,E,E) = sup{K(t,x) : ∥x∥E = 1} ρ(t)=ρ(t,E,E)=inf{J(t,x):x∈E0 ∩E1, ∥x∥E =1}.
It is easy to prove that ψ(t) and ρ(t) are strictly positive and non-decreasing, while ψ(t)/t and ρ(t)/t are non-increasing.
An intermediate space E with respect to E = (E0,E1) is said to be of class CK(θ,E) if there is a constant C such that ψ(t) ≤ Ctθ for every t > 0 and is said of class CJ(θ,E) if there is a constant C such that ρ(t) ≥ Ctθ for every t > 0.
3. s-numbers
A map s that assigns to every operator T acting on Banach spaces a bounded sequence (sn(T)) of real numbers is called an s-function if for all Banach spaces E, E0, F and F0,
(i) ∥T∥=s1(T)≥s2(T)≥...≥0forallT ∈L(E,F);
(ii) sk+n−1(S+T)≤sk(S)+sn(T)forallS,T ∈L(E,F)andk,n∈N;
(iii) sn(R ◦ S ◦ T) ≤ ∥R∥sn(S)∥T∥ for all T ∈ L(E0,E), S ∈ L(E,F), R ∈ L (F, F0) and n ∈ N;
(iv) ifrankT <n,thensn(T)=0,n∈N;
(v) sn(Id:ln2 →ln2)=1foralln∈N.
Here ln2 is the usual euclidean space. We call sn(T) the nth s-number of T.
We shall now give examples of s-numbers. Let E and F be Banach spaces and denote by L(E,F) the space of all bounded linear operator from E into F. Let
T ∈ L (E, F ). The n-th approximation number of T is defined by an(T)=inf{∥T−A∥:A∈L(E,F), rankA<n}.
The nth Kolmogorov number is given by
dn(T ) = inf ∥QFN T ∥ : N is a subspace of F, dim N < n .
Here QFN denotes the canonical surjection from the Banach space F onto the quo- tient space F/N. The nth Gelfand number is given by
cn(T) = inf ∥TJME ∥ : M is a closed subspace of E, codim M < n,