Remarks on the Riesz-Kantorovich formula

TL;DR

This paper investigates the conditions under which the Riesz-Kantorovich formula accurately represents the supremum of two operators in ordered linear spaces, enhancing understanding of lattice and decomposition properties of operator spaces.

Contribution

It identifies sufficient conditions for the formula's validity, extending known results on lattice and decomposition properties of order bounded operator spaces.

Findings

Established new sufficient conditions for the Riesz-Kantorovich formula

Extended characterizations of lattice properties in operator spaces

Provided insights into decomposition properties of order bounded operators

Abstract

The Riesz-Kantorovich formula expresses (under certain assumptions) the supremum of two operators $S, T : X \to Y$ where $X$ and $Y$ are ordered linear spaces as $$ S \vee T (x) = \sup_{0 \leqslant y \leqslant x} [S (y) + T (x - y)]. $$ We explore some conditions that are sufficient for its validity, which enables us to get extensions of known results characterizing lattice and decomposition properties of certain spaces of order bounded linear operators between $X$ and $Y$.