Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach

Abstract

We introduce a new concept of unbounded solutions to the operator Riccati equation $A_{1}X-X{A}_0-XVX+V^{\ast }=0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B}=\left(\begin{array}{ccc}{A}_0& V{V}^{\ast }& A_1\end{array}\right)$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.