The authors study the matrix equation , where is an positive definite matrix, is positive semidefinite, is (arbitrary) and is , block diagonal, with diagonal blocks equal to . The authors impose the condition (i.e. the matrix is positive definite) and prove the existence and uniqueness of the solution in a certain class of positive definite matrices.
These solutions are important in a problem from optimal interpolation theory, see L. A. Sakhnovich [Interpolation theory and its Applications. (Mathematics and Its applications. (Dordrecht). 428 Dordrecht: Kluwer Academic Publishers.) (1997; Zbl 0894.41001), Chapter 7], where existence and uniqueness of the solutions is conjectured.