To the Theory of Operators that are Bounded on Cones in Weighted Spaces of Numerical Sequences

Abstract

The paper is devoted to the general problem of obtaining interpolation theorems for operators that are bounded on cones in normed spaces and to some specific results pertaining to the particular problem of interpolation of operators that are bounded on cones in weighted spaces of numerical sequences. This setting is a natural generalization of the classical problem of interpolation of the boundedness property for a linear operator that is bounded between two Banach couples. We introduce the general concept of a Banach triple of cones possessing the interpolation property with respect to a given Banach triple. We provide sufficient conditions under which a triple of cones (Q 0 , Q 1 , Q) in weighted spaces of numerical sequences possesses the interpolation property with respect to a given Banach triple of weighted spaces of numerical sequences (F 0 , F 1 , F ). Appropriate interpolation theorems generalize the classical result about interpolation of linear operators in weighted spaces and are of interest for the theory of bases in Fréchet spaces. Bibliography: 10 titles.