Page 34 - Textos de Matemática Vol. 34

24 ANTO´NIO J. G. BENTO
where JME is the canonical injection from the subspace M into the Banach space E. Together with these numbers the nth Weyl number
xn(T) = sup{an(TR) : R ∈ L(l2,E),∥R∥ ≤ 1} and the nth Chang number
yn(T) = sup{an(ST) : S ∈ L(F,l2),∥S∥ ≤ 1}
are also s-numbers. Here l2 is the usual space of bounded scalar sequences (xn) ∞
such that |xn|2 is convergent endowed with the usual norm. n=1
The following facts it will be useful
(3.1) yn(T) = sup{dn(ST) : S ∈ L(F,l2),∥S∥ ≤ 1} and
(3.2) xn(T) = sup{cn(TR) : R ∈ L(l2,E),∥R∥ ≤ 1}.
In 1970 Triebel [15] proved the following interpolation theorem.
Theorem 3.1. Let E = (E0,E1) and F = (F0,F1) be Banach couples and T ∈
L E,F.
(i) If F0 = F1 = F and E is an intermediate space of class CK (θ, E), then there
is a constant c such that
dk+n−1(TE,F ) ≤ c d1−θ(TE0,F ) dθn(TE1,F ).
k
(ii) If E0 = E1 = E and F is an intermediate space of class CJ (θ, F ), then there
is a constant c such that
ck+n−1(TE,F ) ≤ c c1−θ(TE,F0 ) cθn(TE,F1 ).
k
Using Triebel’s results and equalities (3.1) and (3.2) Martins [10] proved the following result.
Theorem 3.2. Let E = (E0,E1) and F = (F0,F1) be Banach couples and T ∈ L E,F.
(i) If F0 = F1 = F and E is an intermediate space of class CK (θ, E), then there is a constant c such that
yk+n−1(TE,F ) ≤ c y1−θ(TE0,F ) ynθ (TE1,F ). k
(ii) If E0 = E1 = E and F is an intermediate space of class CJ (θ, F ), then there is a constant c such that
xk+n−1(TE,F ) ≤ c x1−θ(TE,F0 ) xθn(TE,F1 ). k