On the size of certain subsets of invariant Banach sequence spaces

TL;DR

This paper extends methods for identifying large linear structures within invariant Banach sequence spaces, demonstrating that certain subsets contain infinite-dimensional subspaces, thus advancing understanding of their linearity properties.

Contribution

It generalizes recent results on invariant sequence spaces, providing new techniques to find closed infinite-dimensional subspaces within specific subsets.

Findings

Certain subsets contain closed infinite-dimensional subspaces

Extension of previous results to more general invariant sequence spaces

Provides a framework for identifying linear structures in chaotic environments

Abstract

The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use \textit{ad hoc} arguments and few general techniques are known. Motivated by the search of general methods, in this paper we formally extend recent results of G.\ Botelho and V.V. F\'{a}varo on invariant sequence spaces to a more general setting. Our main results show that some subsets of invariant sequence spaces contain, up to the null vector, a closed infinite-dimensional subspace.