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

In \cite{Positive semidefinite solutions}, T. Furuta discusses the existence of positive semidefinite solutions of the operator equation ${\sum }_{j=1}^n{A}^{n-j}X{A}^{j-1}=B$. 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}.