Page 32 - Textos de Matemática Vol. 34

22 ANTO´NIO J. G. BENTO
In this paper we are going to present some interpolation results that we proved in [2] for Kolmogorov, Chang, Gelfand and Weyl numbers of operators acting on general intermediate spaces. These are similar to the results obtained by Matos in [12] for entropy numbers. The basic facts of interpolation theory are given in section 2 and in section 3 we define and enunciate some properties of s-numbers. In sections 4 and 5 we present the interpolation results from Bento [2].
2. Interpolation theory
In this section we give the basic definitions of interpolation theory. For details about this we recommend Bergh and L¨ofstr¨om [3], Triebel [16] and Bennett and Sharpley [1].
A pair E = (E0,E1) of Banach spaces E0 and E1 is called a Banach couple if E0 and E1 are continuously embedded in some Hausdorff topological vector space. Then∆E=E0∩E1 andΣE=E0+E1 areBanachspaceswiththenorms
∥x∥ = max ∥x∥ ∆(E)
E0
, ∥x∥ E1
and respectively.
∥x∥Σ(E)=inf∥x0∥E0 +∥x1∥E1 :x=x0+x1,xi ∈Ei,i=0,1, For each t > 0, we define
J(t,x) = J(t,x,E) = max∥x∥
E0 E1
for every x ∈ ∆E and
K(t, x) = K(t, x, E)
,
,t∥x∥
= inf∥x0∥E0 +t∥x1∥E1 :x=x0 +x1, xi ∈Ei,i=0,1 for every x ∈ Σ E. Obviously,
∥x∥Σ(E) = K(1, x) and ∥x∥∆(E) = J(1, x).
These functionals are called Peetre’s J-functional and Peetre’s K-functional, re- spectively. Moreover, for every t > 0, K(t, ·) and J(t, ·) are norms on Σ E and ∆E equivalent to ∥·∥Σ(E) and ∥·∥∆(E), respectively.
A Banach space E is said to be an intermediate space with respect to a Banach couple E = (E0, E1) if
∆E → E → ΣE,
where → means continuous inclusion. Obviously, ∆ E and Σ E are intermediate spaces with respect to E.
Let F = (F0, F1) be another Banach couple. We denote by L (E, F ) the class of all linear operators T : E0 +E1 → F0 +F1 such that the restriction of T to Ei is a