Dichotomous Hamiltonians with Unbounded Entries and Solutions of Riccati Equations

TL;DR

This paper investigates operator Riccati equations with unbounded Hamiltonian entries, establishing existence, boundedness, and uniqueness of solutions through a dichotomy approach and symmetry properties.

Contribution

It introduces a novel method using a dichotomy property and symmetry to analyze Riccati equations with unbounded Hamiltonian entries, extending previous bounded-entry results.

Findings

Existence of nonnegative and nonpositive solutions is proven.

Conditions for the boundedness of solutions are established.

Uniqueness criteria for solutions are provided.

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.