Quantitative approach to weak noncompactness in the polygon interpolation method

Abstract

We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach TV-tuples. In the case of operators acting between two J-spaces or two if-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.

Measurability and semi-continuity of multifunctions

“…28,29 Measures of noncompactness or weak noncompactness have been successfully applied to the study of compactness, operator theory, differential equations and integral equations, see for instance. 26,27,[29][30][31][32][33][34][35][36][37][38][39] Theorem 2.6 tells us that all classical approaches used so far to study weak compactness in Banach spaces (Tychonoff 's theorem, Eberlein-Šmulian's theorem, Eberlein-Grothendieck double-limit criterion) are qualitatively and quantitatively equivalent. Quantitative versions of James compactness theorem can be found in.…”

A quantitative version of James's Compactness Theorem

“…Such quantities are called measures of weak non-compactness. Measures of non-compactness or weak noncompactness have been successfully applied to study of compactness, in operator theory, differential equations and integral equations, see for instance [2,3,5,6,7,8,9,10,12,13]. An axiomatic approach to measures of weak non-compactness may be found in [4,13].…”