Geometric spaceability in sequence classes and operator ideals

TL;DR

This paper develops a unified framework for sequence spaces and operator ideals to establish advanced spaceability results, extending previous work to non-locally convex settings and higher-dimensional cases.

Contribution

It introduces Standard Sequence Classes and provides new spaceability criteria for complements of unions of sequence spaces and differences of operator ideals.

Findings

Established $(eta, rak c)$-spaceability for complements of unions of sequence spaces.

Provided criteria for pointwise $rak c$-spaceability of differences of operator ideals.

Extended spaceability results to non-locally convex and higher-dimensional cases.

Abstract

This paper investigates advanced notions of lineability and spaceability within the frameworks of sequence spaces and operator ideals. We propose the notion of \emph{Standard Sequence Classes} to provide an environment that unifies numerous classical sequence spaces while preserving their fundamental behavior. Utilizing this framework, we establish general $(\alpha, \mathfrak{c})$-spaceability results for complements of unions of (quasi-)Banach sequence spaces. These results extend the existing literature by addressing the geometrically more demanding case where $\alpha > 1$ and by encompassing the non-locally convex (quasi-)Banach setting. Furthermore, we provide criteria for the pointwise $\mathfrak{c}$-spaceability of differences between general operator ideals with values in standard sequence spaces. Our results recover and improve several known findings in the context of vector-valued sequences.