Page 35 - Textos de Matemática Vol. 34

INTERPOLATION THEORY AND S-NUMBERS 25
A function s that verifies condition (i), (ii) and (iii) of the definition of an s- function is called a pseudo-s-function. The (dyadic) entropy numbers (en(T))n∈N defined by
 n−1  2
(3.3) en(T)=inf ε>0:∃y1,y2,...,y2n−1 ∈F,T(UE)⊆ {yi +εUF}  j=1
form a pseudo-s-function, but do not form an s-function. Here UE and UF are the closed unit balls of E and F, respectively.
The entropy numbers and the Kolmogorov numbers are related by the following relation
(3.4) lim en(T)= lim dn(T). n→∞ n→∞
This quantity is usually represented by β(T ) and called measure of non-compactness of T.
For more details about s-numbers we refer Carl and Stephani [4], Edmunds and Evans [7], K¨onig [9] and Pietsch [13, 14].
4. Interpolation of Kolmogorov and Chang numbers
In this section all the results are from Bento [2, section 5.3.1]. We start with an
interpolation theorem for Kolmogorov numbers that generalises Theorem 3.1/(i).
Theorem 4.1. Assume that E = (E0,E1) is a Banach couple, that E is an intermediate space with respect to E and that F is another Banach space. Let T ∈L(E,F).
(i) If dn(TE1,F ) = 0, then
dk+n−1(TE,F ) ≤ dk(TE0,F ) lim ψ(t, E, E).
t→0
(ii) If dk(TE0,F ) = 0, then
dk+n−1(TE,F)≤dn(TE1,F) lim ψ(t,E,E)/t.
t→∞ (iii) If dk(TE0,F ) > 0 and dn(TE1,F ) > 0, then
dn(TE1,F) dk+n−1(TE,F)≤2dk(TE0,F)ψ dk(TE0,F),E,E .
The main step of the proof is the following inequality
(4.1) dk+n−1(TE,F ) ≤ dk(TE0,F )ψ(t) + dn(TE1,F )ψ(t)/t
for every t > 0 and every k,n ∈ N. From this inequality we also obtain the following theorem that was proved by Cobos, Cwikel and Matos [5, Theorem 3.1] using a different approach.