Characterizations of operator order for k strictly positive operators

TL;DR

This paper provides new characterizations of the operator order among multiple positive operators using inequalities and applies these results to operator equalities based on Douglas's majorization and factorization theorem.

Contribution

It introduces novel inequality-based characterizations of operator order for multiple positive operators and connects these to operator equalities via Douglas's theorem.

Findings

Characterizations of operator order using inequalities

Application to operator equalities

Extension of Douglas's majorization theorem

Abstract

Let $A_{i}\ (i=1, 2, ..., k)$ be bounded linear operators on a Hilbert space. This paper aims to show characterizations of operator order $A_{k}\geq A_{k-1}\geq...\geq A_{2}\geq A_{1}>0$ in terms of operator inequalities. Afterwards, an application of the characterizations is given to operator equalities due to Douglas's majorization and factorization theorem.