On certain higher order Riccati-type operator equations with possibly unbounded operator coefficients (Q1100704)

The author proves the existence of a solution \(X\in L(V\) 2,V 1) of the operator equation \[ (*)\quad A_ 1XA_ 2-B_ 1XB_ 2+XDX+XEXFX=Q. \] Let H and V 1 be complex Hilbert spaces, with V 1 topologically included in H, and V 2 a complex pre-Hilbert space. Let \(A_ 1\), \(B_ 1\in L(V\) 1,H),D,E,F\(\in L(V\) 1,V 2), \(Q\in L(V\) 2,H) with one-dimensional range and \(A_ 2,B_ 2:V\) \(2\to V\) 1 linear. Under some hypothesis, an iteration method is used to prove the existence of a solution of (*). The proof is based on a theorem of the author [Z. Angew. Math. Mech. 66, 443- 444 (1986; Zbl 0613.47014)] for the equation (*) with \(D=E=F=0\). A finite-dimensional and an infinite-dimensional example are given.