Positive definite solution of the matrix equation \(X=Q+A^{H}(I\otimes X-C)^{\delta}A\) (Q633150)

The authors consider the nonlinear matrix equation \(X=Q+A^H(I \otimes X - C)^{\delta}A\), \((\delta = -1\) or \(0 < |\delta| < 1)\), where \(Q\) is an \(n \times n\) positive definite matrix, \(C\) is an \(mn \times mn\) positive semidefinite matrix, \(I\) is the \(m \times m\) identity matrix, and \(A\) is an arbitrary \(mn \times n\) matrix. Under the assumption that \(I \otimes Q > C\), they prove the existence and uniqueness of its positive definite solution which is contained in the set \(\{ X \mid I \otimes X > C \}\).