Tauberian pairs of closed subspaces of a Banach space

TL;DR

This paper introduces new concepts of tauberian, cotauberian, and weakly compact pairs of closed subspaces in Banach spaces, expanding the theoretical framework beyond operators.

Contribution

It develops a richer theory for pairs of subspaces, extending operator concepts to closed subspaces and exploring applications in Banach space indecomposability.

Findings

Defined new classes of subspace pairs: tauberian, cotauberian, weakly compact.

Extended operator notions to pairs of closed subspaces.

Applied these notions to analyze Banach space structure.

Abstract

We introduce the notions of tauberian, cotauberian and weakly compact pair of closed subspaces of a Banach space. The theory produced by these notions is richer than that of the corresponding operators since an operator can be regarded as a suitable pair of closed subspaces. We investigate into these classes of pairs of subspaces and describe several applications in order to define some notions of indecomposability for Banach spaces and in order to extend definitions from the case of bounded operators to the case of closed operators.