Page 37 - Textos de Matemática Vol. 34

INTERPOLATION THEORY AND S-NUMBERS
(ii) If dn(T∆(E),F ) ̸= 0, then
dn(T∆(E),F)
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∥T∥ E,F
dn(T∆(E),F )
dn(TE,F)≤4max ∥T∥E,F ψ
,dn(T∆(E),F)ψ In the proof of this theorem we got the following inequality
∥T ∥E,F
where ψ(t) = ψ(t, E, E).
(4.2) dn(TE,F ) ≤ 2 ∥T ∥E,F η(t) + dn(T∆(E),F )η(t)(1 + 1/t)
that is true for every t > 0 and every n ∈ N. From this inequality and (3.4) we have the following result that was proved in a more general setting by Cobos, Manzano, Martin´ez and Matos [6, Theorem 3.3] using a different approach.
Theorem 4.5. Let E = (E0,E1) be a Banach couple, let E be an intermediate space with respect to E, let F be a Banach space and assume that T ∈ L (E, F ).
(i) If β(T∆(E),F ) = 0, then
β(TE,F ) ≤ 2 ∥T ∥E,F max lim ψ(t, E, E), lim ψ(t, E, E)/t .
(ii) If β(T∆(E),F ) ̸= 0, then
β(T∆(E),F) ∥T ∥E,F
∥T∥ E,F
β(T∆(E),F )
t→0 t→∞
β(TE,F)≤4max ∥T∥E,F ψ
,β(T∆(E),F)ψ From (3.1) and (4.2) we obtain the following result.
where ψ(t) = ψ(t, E, E).
Theorem 4.6. Let E = (E0,E1) be a Banach couple, let E be an intermediate space with respect to E, let F be a Banach space and assume that T ∈ L (E, F ).
(i) If yn(T∆(E),F ) = 0, then
yn(TE,F)≤2∥T∥E,F maxlimψ(t,E,E), lim ψ(t,E,E)/t.
(ii) If yn(T∆(E),F ) ̸= 0, then
yn(T∆(E),F)
∥T∥ E,F
yn(T∆(E),F )
yn(TE,F)≤4max ∥T∥E,F ψ where ψ(t) = ψ(t, E, E).
∥T ∥E,F
,yn(T∆(E),F)ψ
t→0 t→∞