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Tauberian pairs of closed subspaces of a Banach space

Manuel González, Antonio Martínez-Abejón

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.

Tauberian pairs of closed subspaces of a Banach space

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.

Paper Structure

This paper contains 7 sections, 20 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tauberian, cotauberian and weakly compact pairs
  4. Some applications and questions
  5. Incomparability of Banach spaces in terms of $\mathcal{W}_+$ and $\mathcal{W}_-$ pairs
  6. Classes of closed operators
  7. Some questions

Key Result

Proposition 3.2

GO:90KW:76 For $S\in\mathcal{B}(X,Y)$, the following assertions are equivalent:

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Lemma 3.8
  • ...and 15 more