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

  • Tapas Mazumdar

    Wright State University, Dayton, USA

Abstract

Let H\mathcal H and V1\mathcal V^1 be complex Hilbert spaces, with V1\mathcal V^1 topologically included in H\mathcal H, and V2\mathcal V^2 a complex pre-Hilbert space. There is considered the existence of a solution X=XQL(V2,V1)X = X_Q \in \mathcal L (\mathcal V^2, \mathcal V^1) of the operator equation

A1XA2B1XB2+XDX+XEXFX=QA_1XA_2 — B_1XB_2 +XDX + XEXFX = Q

in the space of (bounded or not) linear operators in H\mathcal H under the data A1,B1L(V1,H),D,E,FL(V1,V2);A2,B2:V2V2A_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 QL(V2,H)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.

Cite this article

Tapas Mazumdar, On Certain Higher Order Riccati-Type Operator Equations with Possibly Unbounded Operator Coefficients. Z. Anal. Anwend. 6 (1987), no. 5, pp. 409–419

DOI 10.4171/ZAA/260