Dichotomous Hamiltonians with Unbounded Entries and Solutions of Riccati Equations

Abstract

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.

31 pages, 3 figures; proof of uniqueness of solutions added; to appear in Journal of Evolution Equations

References in corpus (2)

• Riesz Bases for p-Subordinate Perturbations of Normal Operators
• Hamiltonians with Riesz Bases of Generalised Eigenvectors and Riccati Equations

Cited by in corpus (6)

• On the Spectral Decomposition of Dichotomous and Bisectorial Operators
• Pseudo Numerical Ranges and Spectral Enclosures
• On invariant graph subspaces
• On symplectic self-adjointness of Hamiltonian operator matrices
• Relative residual bounds for eigenvalues in gaps of the essential spectrum
• On invertible nonnegative Hamiltonian operator matrices