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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 (∗).

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