On Certain Higher Order Riccati-Type Operator Equations with Possibly Unbounded Operator Coefficients

Abstract

Let $\mathcal{H}$ and ${\mathcal{V}}^1$ be complex Hilbert spaces, with ${\mathcal{V}}^1$ topologically included in $\mathcal{H}$, and ${\mathcal{V}}^2$ a complex pre-Hilbert space. There is considered the existence of a solution $X=X_Q\in \mathcal{L}({\mathcal{V}}^2,{\mathcal{V}}^1)$ of the operator equation

in the space of (bounded or not) linear operators in $\mathcal{H}$ under the data $A_1,B_1\in \mathcal{L}({\mathcal{V}}^1,\mathcal{H}),D,E,F\in \mathcal{L}({\mathcal{V}}^1,{\mathcal{V}}^2);A_2,B_2:{\mathcal{V}}^2\to {\mathcal{V}}^2$ linear and $Q\in \mathcal{L}({\mathcal{V}}^2,\mathcal{H})$ one-dimensional. Under some hypotheses, an iterative analytic metiod to arrive at the existence of such a solution is given. Two examples are given.