On Hermitian positive definite solutions of matrix equation X+A*X−2A=I

The Hermitian positive definite solutions of the matrix equation X+A*X−2A=I are studied. A necessary and sufficient condition for existence of solutions is given in case A is normal. The basic fixed point iterations for the equation in case A is nonnormal with are discussed in some detail. Some of Ivanov’s, Hasanov’s and Minchev’s results in [Linear Algebra Appl. 326 (2001) 27] are improved.