Duality between Y-convexity and $Y^{\times}$-concavity of linear operators between Banach lattices

TL;DR

This paper explores the relationship between Y-convexity and Y^{×}-concavity in linear operators between Banach lattices, introducing new vector sequence spaces and establishing a duality result.

Contribution

It introduces a characterization of Y-convexity and Y-concavity via vector sequence spaces and proves a duality between these properties in Banach lattice operators.

Findings

Characterization of Y-convexity using vector sequence spaces

Analogous results for Y-concavity

Duality between Y-convexity and Y^{×}-concavity

Abstract

In this paper we study the Y-convexity, a property which is obtained by considering a real Banach sequence lattice Y instead of $\ell^p$ for a linear operator $T : E \rightarrow X$, where E is a Banach space and X is a Banach lattice. We introduce some vector sequence spaces in order to characterize the Y-convexity of T by means of the continuity of an associated operator $\overline{T}$. Analogous results for Y-concavity are also obtained. Finally, the duality between Y-convexity and $Y^{\times}$-concavity is proven.