On Pairs of l-Köthe Spaces

Abstract

Let ℓ be a Banach sequence space with a monotone norm k · kℓ, inwhich the canonical system (ei) is a normalized unconditional basis.Let a = (ai), ai → ∞, λ = (λi) be sequences of positive numbers. Westudy the problem on isomorphic classification of pairsF =Kℓexp −1pai, Kℓexp −1pai + λi.For this purpose, we consider the sequence of so-called m-rectanglecharacteristics µFm. It is shown that the system of all these characteristics is a complete quasidiagonal invariant on the class of pairs offinite-type ℓ-power series spaces. By using analytic scale and a modification of some invariants (modified compound invariants) it is proventhat m-rectangular characteristics are invariant on the class of suchpairs. Deriving the characteristic βe from the characteristic β, and using the interpolation method of analytic scale, we are able to generalizesome results of Chalov, Dragilev, and Zahariuta (Pair of finite typepower series spaces, Note di Mathematica 17, 121–142, 1997).