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

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

2002-07-15

arxiv.org/abs/math/0207125v1

Contemporary Mathematics Vol. 327, Amer. Math. Soc., 2003, p. 181 - 198

Mathematics

Spectral Theory

Scientific paper

We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. 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.

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