Further development of positive semidefinite solutions of the operator equation $\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$

TL;DR

This paper advances the understanding of positive semidefinite solutions for a specific operator equation by applying the Grand Furuta inequality, leading to a generalized form of the operator B.

Contribution

It introduces a new approach using the Grand Furuta inequality to analyze and generalize solutions of the operator equation involving positive semidefinite operators.

Findings

Derived a generalized form of B for the operator equation

Applied the Grand Furuta inequality to obtain new solutions

Extended previous results on positive semidefinite solutions

Abstract

In \cite{Positive semidefinite solutions}, T. Furuta discusses the existence of positive semidefinite solutions of the operator equation $\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$. In this paper, we shall apply Grand Furuta inequality to study the operator equation. A generalized special type of $B$ is obtained due to \cite{Positive semidefinite solutions}.