Geometric spaceability in sequence classes and operator ideals

TL;DR

The paper introduces Standard Sequence Classes to unify vector-valued sequence spaces and develops general $(\alpha,\mathfrak{c})$-spaceability results for complements of unions of (quasi-)Banach sequence spaces, including the non-locally convex quasi-Banach setting and cases with $\alpha$ finite. It also establishes detachable triples to obtain sharp criteria for the pointwise spaceability of differences between operator ideals valued in standard sequence spaces, and provides a main geometric-spaceability theorem plus a parallel operator-ideals theorem followed by broad applications to both standard sequence spaces and operator ideals. The results extend and refine existing lineability/spaceability findings for vector-valued sequences and furnish versatile tools for constructing large linear structures inside non-linear sets. Collectively, the work broadens the scope of spaceability in functional-analytic settings, offering unified techniques and concrete corollaries that recover prior results while addressing previously open regimes.

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 $(α, \mathfrak{c})$-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 $α> 1$ 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.

Theorems & Definitions (30)

• Definition 2.1
• Example 2.2
• Proposition 2.3
• proof
• Definition 2.4
• Remark 2.5
• Example 2.6
• Proposition 2.7
• Proposition 2.8
• proof