On Pairs of ℓ-Köthe Spaces

Abstract

Let ℓ be a Banach sequence space with a monotone norm {double pipe}· {double pipe} ℓ, in which the canonical system (ei) is a normalized unconditional basis. Let a = (ai), ai → ∞, λ = (λi) be sequences of positive numbers. We study the problem on isomorphic classification of pairs F = (Kℓ(exp(-1/p ai)),Kℓ(exp (-1/p ai + λi))). For this purpose, we consider the sequence of so-called m-rectangle characteristics μF m. It is shown that the system of all these characteristics is a complete quasidiagonal invariant on the class of pairs of finite-type ℓ-power series spaces. By using analytic scale and a modification of some invariants (modified compound invariants) it is proven that m-rectangular characteristics are invariant on the class of such pairs. Deriving the characteristic β̃ from the characteristic β, and using the interpolation method of analytic scale, we are able to generalize some results of Chalov, Dragilev, and Zahariuta (Pair of finite type power series spaces, Note di Mathematica 17, 121-142, 1997).