Operator inequalities dealing with operator equations

TL;DR

This paper explores solutions to operator equations using operator inequalities, characterizing operator order and applying Furuta inequality to establish solution criteria for various operator equations.

Contribution

It introduces new characterizations of operator order and chaotic operator order via operator equations, extending previous results and applying Furuta inequality in this context.

Findings

Characterizations of operator order and chaotic operator order in terms of operator equations

Generalization of previous results on operator inequalities

Application of Furuta inequality to solve operator equations

Abstract

In this paper, we study the existence of solutions of some kinds of operator equations via operator inequalities. First, we investigate characterizations of operator order $A\geqslant B >0$ and chaotic operator order log $A \geqslant$ log $B$ for positive definite operators $A$, $B$ in terms of operator equations, and generalize the results in \cite{CSLin}. Then, we introduce applications of complete form of Furuta inequality in operator equations. Some kinds of operator equations are researched and related characterizations of solutions are proved.