Page 31 - Textos de Matemática Vol. 34

INTERPOLATION THEORY AND S-NUMBERS ANTO´NIO J. G. BENTO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. We present some interpolation inequalities of s-numbers of opera- tors acting on general intermediate spaces
1. Introduction
In the last twenty five years estimates of s-numbers and entropy numbers of embeddings between function spaces have been used to study the asymptotical behaviour of eigenvalues of (pseudo-)differential operators and integral operators (see [9], [8], [17] and [18], for example). A useful tool to obtain these estimates is interpolation theory.
The first interpolation results for s-numbers of operators were obtained in 1970 by Triebel [15] (see also Pietsch [13]). Triebel proved interpolation inequalities for Kolmogorov and Gelfand numbers of operators. In the same paper, Triebel also obtained interpolation results for the ε-entropy of operators. The first interpolation results for entropy numbers of operators were presented, as far as we are aware, by Pietsch in [13]. Similar results for Weyl and Chang numbers were proved by Martins in [10] (see also [11]).
All the results mention so far are for operators acting on special classes of in- termediate spaces. For Kolmogorov and Chang numbers the intermediate space is of class CK(θ) and for Gelfand and Weyl numbers is of class CJ(θ). For entropy numbers in one of the cases the intermediate space is of class CK(θ) and in the other case is of class CJ (θ).
In 1999 Matos [12] proved interpolation results for entropy numbers of operators without imposing any condition on the intermediate spaces. For that he used the functions ψ and ρ introduced by Cobos, Cwikel and Matos in [5] to interpolate the measure of non-compactness without any assumption on the intermediate spaces.
2000 Mathematics Subject Classification. 46B70, 47B06.
Key words and phrases. s-numbers, interpolation theory, intermediate spaces.
This work was supported by Centro de Matem´atica da Universidade da Beira Interior.
21