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 (Q2276734)

The paper treats the operator equation \[ (*)\quad XA+BX=C, \] where all operators are bounded linear ones on a Hilbert space, and the spectra \(\sigma\) (A) and \(\sigma\) (-B) are disjoint. Then the equation (*) has a unique solution X. In this paper the authors find effectively both a polynomial p and an operator W such that the solution X of (*) can be obtained by a computer- friendly equation \[ (\#)\quad X-p(-B)Xp(A)^{-1}=W, \] under the two assumptions (1) A is a normal operator and (2) \(\sigma\) (A)\(\cup \sigma (-B)\) has no holes in the complex plane.