Spaceability in Banach and quasi-Banach sequence spaces

TL;DR

This paper investigates the structure of certain sequence spaces in Banach and quasi-Banach settings, showing they contain large closed subspaces and applying these results to spaces of norm-attaining operators.

Contribution

It introduces new techniques to identify large closed subspaces within Banach and quasi-Banach sequence spaces, extending previous results and improving understanding of norm-attaining operators.

Findings

Sets like E minus unions of ℓ_q spaces contain large closed subspaces.

The same techniques improve results on spaces of norm-attaining operators.

Applicable to a broad class of Banach and quasi-Banach spaces.

Abstract

Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain closed infinite-dimensional subspaces of $E$ (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.