Page 39 - Textos de Matemática Vol. 34

INTERPOLATION THEORY AND S-NUMBERS 29
The last result was proved by Cobos, Cwikel and Matos [5, Theorem 4.2] in a more general setting and using a different approach.
From (3.2) we have the following result that generalises Theorem 3.2/(ii).
Theorem 5.3. Let F = (F0,F1) be a Banach couple, let F be an intermediate space with respect to F, let E be a Banach space and assume that T ∈ L(E,F).
(i) If xn(TE,F1 ) = 0, then
xk+n−1(TE,F ) ≤ xk(TE,F0 ) lim 1
t→∞ ρ(t,F,F)
.
(ii) If xk(TE,F0 ) = 0, then
xk+n−1(TE,F ) ≤ xn(TE,F1 ) lim t
t→0 ρ(t,F,F) xk(TE,F0)
.
(iii) If xk(TE,F0 ) > 0 and xn(TE,F1 ) > 0, then
xk+n−1(TE,F)≤ ρxn(TE,F1)/xk(TE,F0),F,F.
In next theorem we compare the Gelfand numbers of the operators T : E → F with T : E → Σ F .
Theorem 5.4. Let F = (F0,F1) be a Banach couple, let F be an intermediate space with respect to F, let E be a Banach space and assume that T ∈ L(E,F).
(i) If cn(TE,Σ(F )) = 0, then cn(TE,F)≤∥T∥E,F lim
t→0 ρ(t,F,F)
t1
+ lim . t→∞ ρ(t,F,F)
(ii) If cn(TE,Σ(F )) ̸= 0, then 2cn(TE,Σ(F ))
2 ∥T ∥E,F
+ .
c (T )≤ 1 maxc T ,t−1∥T∥ +c T n E,F ρ(t−1) n E,Σ(F) E,F n E,Σ(F)
(5.2) + 1 max∥T∥ +c T ,tc T ρ(t) E,F n E,Σ(F ) n E,Σ(F )
for every t > 0 and every n ∈ N.
In next theorem we replace the Gelfand numbers by their limit. This was proved
by Cobos, Manzano, Mart´ınez and Matos in [6, Theorem 3.1] in a more general setting and using a different approach.
cn(TE,F)≤
ρ cn TE,Σ(F) /∥T∥E,F ,F,F
ρ ∥T∥E,F /cn TE,Σ(F) ,F,F In the proof of this theorem it was necessary to obtain the following inequality