S-Decomposable Banach Lattices, Optimal Sequence Spaces and Interpolation

TL;DR

This paper explores the relationship between Banach lattices, interpolation theory, and sequence spaces, providing characterizations, factorizations, and properties of optimal sequence spaces with applications to Orlicz spaces.

Contribution

It introduces a new characterization of s-decomposable Banach lattices via sequence space estimates and studies their properties, including rearrangement invariance and applications to interpolation.

Findings

Characterization of s-decomposable Banach lattices using sequence space estimates

Orbital factorization of K-functional estimates through weighted Lp-spaces

Optimal upper and lower sequence spaces are rearrangement invariant

Abstract

We investigate connections between upper/lower estimates for Banach lattices and the notion of relative s-decomposability, which has roots in interpolation theory. To get a characterization of relatively s-decomposable Banach lattices in terms of the above estimates, we assign to each Banach lattice X two sequence spaces XU and XL that are largely determined by the set of p, for which lp is finitely lattice representable in X. As an application, we obtain an orbital factorization of relative K-functional estimates for Banach couples (X0, X1) and (Y0, Y1) through some suitable couples of weighted Lp-spaces provided if Xi, Yi are relatively s-decomposable for i = 0, 1. Also, we undertake a detailed study of the properties of optimal upper and lower sequence spaces XU and XL, and, in particular, prove that these spaces are rearrangement invariant. In the Appendix, a description of the optimal upper sequence space for a separable Orlicz space as a certain intersection of some special Musielak-Orlicz sequence spaces is given