Algebras of Lebesgue and KB regular operators in Banach lattices

TL;DR

This paper investigates the algebraic structures formed by Lebesgue and KB regular operators in Banach lattices, establishing conditions under which these operators form Banach lattice algebras and characterizing KB-spaces via positive compact operators.

Contribution

It proves that Lebesgue and KB regular operators form operator algebras in Banach lattices and characterizes KB-spaces through positive compact operators.

Findings

Regular Lebesgue and KB operators form operator algebras in Banach lattices.

In Dedekind complete lattices, these operators form Banach lattice algebras.

A Banach lattice with order continuous norm is a KB-space iff all positive compact operators are KB operators.

Abstract

It is shown that the regularly Lebesgue and regularly (quasi) KB operators in a Banach lattice E form operator algebras and, for Dedekind complete E, even Banach lattice algebras. We also prove that a Banach lattice E with order continuous norm is a KB-space iff each positive compact operator in E is a KB operator.