A sufficient condition on operator order for strictly positive operators

TL;DR

This paper establishes a sufficient condition for a chain of strictly positive operators on a Hilbert space to be ordered, extending previous results in operator theory.

Contribution

It introduces a new sufficient condition for the operator order among multiple strictly positive operators, broadening existing theoretical frameworks.

Findings

Provides a generalized condition for operator order

Extends previous results in operator inequalities

Enhances understanding of positive operator sequences

Abstract

Let $A_{1}$, $A_{2}$, $...$, $A_{k}$ be strictly positive operators on a Hilbert space. This note is to show a sufficient condition of $A_{k}\geq A_{k-1}\geq\geq A_{3}\geq A_{2}\geq A_{1}$, which extends the related result before.