CO-ISOMETRIC SOLUTIONS OF EQUATION CU + U*C = 2D

This note discusses the co-isometric solutions of the operator equation CU + U*C = 2D, establishes a correspondence between such solutions and the self-adjoint solutions of the algebraic Riccati equation X 2 - i D X + i X D + D 2 + C 2 = 0 , and gives all possible co-isometric solutions parametrically. Some mistakes of Dobovivsek’s results are corrected.