Page 40 - Textos de Matemática Vol. 34

30 ANTO´NIO J. G. BENTO
Theorem 5.5. 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 c(TE,Σ(F )) = 0, then c(TE,F)≤∥T∥E,F lim
t→0 ρ(t,F,F)
t1
+ lim . t→∞ ρ(t,F,F)
(ii) If c(TE,Σ(F )) ̸= 0, then 2c(TE,Σ(F ))
2 ∥T ∥E,F
+ .
Theorem 5.6. 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).
c(TE,F)≤
ρ c TE,Σ(F) /∥T∥E,F ,F,F
ρ ∥T∥E,F /c TE,Σ(F) ,F,F We finish this paper with a theorem for Weyl numbers.
(i) If xn(TE,Σ(F )) = 0, then xn(TE,F)≤∥T∥E,F lim
t→0 ρ(t,F,F)
t1
+ lim . t→∞ ρ(t,F,F)
(ii) If xn(TE,Σ(F )) ̸= 0, then 2xn(TE,Σ(F ))
2 ∥T ∥E,F
+ .
ρ ∥T∥E,F /xn TE,Σ(F) ,F,F
xn(TE,F)≤
ρ xn TE,Σ(F) /∥T∥E,F ,F,F
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