Optimal upper and lower sequence spaces with applications

TL;DR

This paper investigates the construction of optimal upper and lower sequence spaces associated with Banach lattices, revealing their properties and applications in characterizing various function spaces and tensor products.

Contribution

It introduces and analyzes the optimal upper and lower sequence spaces for Banach lattices, providing new characterizations and properties relevant to functional analysis.

Findings

Identified Banach lattices with equal-norm upper and lower p-estimates

Characterized $L_p()$-spaces using these sequence spaces

Analyzed tensor product properties in Lorentz and Orlicz spaces

Abstract

We study the optimal upper $X_U$ and lower $X_L$ sequence spaces that can be assigned to each Banach lattice $X$. These spaces are symmetric, have the Fatou property and the unit vector basis has in these spaces very special properties. Determined by the order structure of $X$ the spaces $X_U$ and $X_L$ turn out to be very useful when studying Banach lattices. Among other results, in terms of these constructions, we identify Banach lattices that satisfy equal-norm upper and lower $p$-estimates, give a characterization of $L_p(\mu)$-spaces, derive some properties of the tensor product operator in Lorentz and Orlicz spaces, identify Orlicz spaces in which the unit vector basis is upper (resp. lower) semi-homogeneous.