Positive semidefinite solutions of the operator equation ∑j=1nAn-jXAj-1=B

Let AA be a positive definite operator and BB be a self-adjoint operator. We discuss the existence of positive semidefinite solutions of the operator equation∑j=1nAn-jXAj-1=Bvia an order preserving operator inequality. By using the solutions we also give concrete examples of positive semidefinite matrices.