Page 38 - Textos de Matemática Vol. 34

28 ANTO´NIO J. G. BENTO
5. Interpolation of Gelfand and Weyl numbers
All the results presented in this section were proved in Bento [2, Section 5.3.2]. These are the dual results of last section. We start with a generalisation of Theorem 3.1/(ii).
Theorem 5.1. 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,F1 ) = 0, then
ck+n−1(TE,F ) ≤ ck(TE,F0 ) lim 1
t→∞ ρ(t,F,F)
.
(ii) If ck(TE,F0 ) = 0, then
ck+n−1(TE,F ) ≤ cn(TE,F1 ) lim t
t→0 ρ(t,F,F)
(iii) If ck(TE,F0 ) > 0 and cn(TE,F1 ) > 0, then ck(TE,F0)
.
ρ(t)
for every t > 0 and every k,n ∈ N was obtained in the proof of Theorem 5.1.
Putting c(T) = lim cn(T) and using last inequality we have the following result. n→∞
Theorem 5.2. 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).
ck+n−1(TE,F)≤ ρcn(TE,F1)/ck(TE,F0),F,F. (5.1) ck+n−1(TE,F ) ≤ max {ck(TE,F0 ), tcn(TE,F1 )}
The inequality
(i) If c(TE,F1 ) = 0, then
c(TE,F ) ≤ c(TE,F0 ) lim 1 t→∞ ρ(t,F,F)
.
(ii) If c(TE,F0 ) = 0, then
c(TE,F ) ≤ c(TE,F1 ) lim t
t→0 ρ(t,F,F)
(iii) Ifc(TE,F0)>0andc(TE,F1)>0,then
.
c(TE,F0 )
c(TE,F ) ≤ ρ c(TE,F1 )/c(TE,F0 ), F, F .