We consider the solution of (*) XA+BX = C for bounded operators A,B,C and X on a Hilbert space, A normal. We establish the existence of a polynomial p and a bounded operator W with the property that the unique solution X of (*) also solves X − p(−B)Xp(A)−1 = W uniquely. A known iterative algorithm can be applied to the latter equation to solve (*).
& Miller, D.
(1990). On the Equivalence of the Operator Equations XA + BX = C and X - p(-B)Xp(A)(-1) = W in a Hilbert-Space, p A Polynomial. Rocky Mountain Journal of Mathematics, 20 (2), 475-486.