Spaceability on some classes of Banach spaces

TL;DR

This paper investigates the spaceability of certain subsets within generalized Orlicz and Lebesgue spaces, providing new conditions for spaceability and demonstrating the existence of large linear structures in operator spaces.

Contribution

It introduces improved sufficient conditions for spaceability in Banach spaces and applies these results to show the spaceability of non-positive semidefinite operators.

Findings

Established new sufficient conditions for spaceability in Banach spaces.

Proved the set of non-positive semidefinite operators is spaceable.

Extended spaceability results to generalized Orlicz and Lebesgue spaces.

Abstract

In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an important result on this topic. As an application, it is shown that the set of all bounded linear operators which are not positive semidefinite on a separable Hilbert space is spaceable.