On (a, B, c)-ideals in Banach spaces

Abstract

In this paper we focus on subspaces of Banach spaces that are (a, B, c)-ideals. We study (a, B, c)-ideals in l²_∞ and present easily verifiable conditions for a subspace of l²_∞ to be an (a, B, c)-ideal. Our main results concern the transitivity of (a, B, c)-ideals. We show that if X is an (a, B, c)-ideal in Y and Y is a (d, E, f)-ideal in Z, then X is a certain type of ideal in Z. Relying on this result, we show that if X is an (a, B, c)-ideal in its bidual, then X is a certain type of ideal in X⁽²ⁿ⁾ for every n ∈ N.