An application of grand Furuta inequality to a type of operator equation

The existence of positive semidefinite solutions ofthe operator equation $\displaystyle\sum_{j=1}^{n}A^{n-j}XA^{j-1}=Y$ is investigated by applying grand Furuta inequality. If there exists positive semidefinite solutions of the operator equation, one of the special types of Y is obtained, which extends the related result before. Finally, an example is given based on our result. Keywords: grand Furuta inequality, operator equation, matrix equation, positive semidefinite operator.