Some investigation on Hermitian positive definite solutions of the matrix equation Xs+A∗X-tA=Q

In this paper, the Hermitian positive definite solutions of the matrix equation Xs+A∗X-tA=Q are considered, where Q is an Hermitian positive definite matrix, s and t are positive integers. Necessary and sufficient conditions for the existence of an Hermitian positive definite solution are derived. A sufficient condition for the equation to have only two different Hermitian positive definite solutions and the formulas for these solutions are obtained. In particular, the equation with the case is discussed. A necessary condition for the existence of an Hermitian positive definite solution and some new properties of the Hermitian positive definite solutions are given, which generalize the existing related results.