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Zeitschrift für Analysis und ihre Anwendungen


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Volume 6, Issue 5, 1987, pp. 409–419
DOI: 10.4171/ZAA/260

Published online: 1987-10-31

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

Tapas Mazumdar[1]

(1) Wright State University, Dayton, USA

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 $$A_1XA_2 — B_1XB_2 +XDX + XEXFX = Q$$ 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.

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Mazumdar Tapas: On Certain Higher Order Riccati-Type Operator Equations with Possibly Unbounded Operator Coefficients. Z. Anal. Anwend. 6 (1987), 409-419. doi: 10.4171/ZAA/260