Dichotomous Hamiltonians with unbounded entries and solutions of Riccati equations. (English) Zbl 1320.47012
The paper under review is devoted to the investigation of algebraic Riccati equations
\[
A^*X + XA + XBX - C =0
\]
on a Hilbert space \(H\), where \(A, B\geq 0, C\geq 0\) are unbounded linear operators. The authors prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Conditions for the boundedness and uniqueness of these solutions are established.
The proof is based on a a certain dichotomy property of the associated Hamiltonian \[ \begin{pmatrix} A&B\\ C&-A^* \end{pmatrix} \] and its symmetry with respect to two different indefinite inner products.
Examples involving partial differential operators are included.
The proof is based on a a certain dichotomy property of the associated Hamiltonian \[ \begin{pmatrix} A&B\\ C&-A^* \end{pmatrix} \] and its symmetry with respect to two different indefinite inner products.
Examples involving partial differential operators are included.
Reviewer: Michael Perelmuter (Kyïv)
MSC:
| 47A62 | Equations involving linear operators, with operator unknowns |
| 47B44 | Linear accretive operators, dissipative operators, etc. |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
Keywords:
Riccati equation; Hamiltonian; dichotomous; bisectorial; invariant subspace; \(p\)-subordinate perturbationReferences:
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