On the Exact K -Monotonicity of Banach Couples

Abstract

We present necessary and sufficient conditions for a Banach couple formed by the space L∞ and an arbitrary Lorentz space Λ(ϕ )t o be exactK-monotone. The proof relies on the description of the set of extreme points of a K-orbit for appropriate finite-dimensional couples. As a consequence of this description, we obtain a generalization of a well-known Markus theorem. i=1 vix ∗ < ∞, where (x ∗ ) is the nonincreasing rearrangement of (|xi|) n=1 . The corresponding Lorentz function space Λ(ϕ )( ϕ is a nonnegative increasing concave function on (0, ∞), ϕ(0) = 0) is formed by the measurable functions x = x(t )o n (0, ∞) such thatx� ϕ = ∞ 0 x ∗ (t) dϕ(t) < ∞ (x ∗ (t) stands for the nonincreasing rearrangement of |x(t)|). As usual, denote by l n (L∞) the space of bounded sequences x =( xi) n=1 (of essentially bounded functions x = x(t)) with the norm