Page 36 - Textos de Matemática Vol. 34

26 ANTO´NIO J. G. BENTO
Theorem 4.2. 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 β(TE1,F ) = 0, then
β(TE,F ) ≤ β(TE0,F ) lim ψ(t, E, E).
t→0
(ii) If β(TE0,F ) = 0, then
β(TE,F)≤β(TE1,F) lim ψ(t,E,E)/t.
t→∞ (iii) If β(TE0,F ) > 0 and β(TE1,F ) > 0, then
β(TE1,F) β(TE,F)≤2β(TE0,F)ψ β(TE0,F),E,E .
Using (3.1) and (4.1) we obtain the following interpolation theorem for Chang numbers. This generalises Theorem 3.2/(i).
Theorem 4.3. 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 yn(TE1,F ) = 0, then
yk+n−1(TE,F ) ≤ yk(TE0,F ) lim ψ(t, E, E).
t→0
(ii) If yk(TE0,F ) = 0, then
yk+n−1(TE,F)≤yn(TE1,F) lim ψ(t,E,E)/t.
t→∞ (iii) If yk(TE0,F ) > 0 and yn(TE1,F ) > 0, then
yn(TE1,F) yk+n−1(TE,F)≤2yk(TE0,F)ψ yk(TE0,F),E,E .
In next theorem we compare the Kolmogorov numbers of the operators T : E → F and T : ∆E → F.
Theorem 4.4. 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 dn(T∆(E),F ) = 0, then
dn(TE,F)≤2∥T∥E,F maxlimψ(t,E,E), lim ψ(t,E,E)/t. t→0 t→∞