The dual algebraic Riccati equations: Additional results under isomorphism of the Riccati operator

In the abstract hyperbolic-like case, under a mild exact controllability assumption, the Riccati operator is known to be an isomorphism [F. Flandoli, I. Lasiecka, R. Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Annali di Matematica Pura e Applicata (iv)CLII (1988) 307-382 (Section 6)]. This property then plays a crucial role in establishing a Dual Algebraic Riccati Theory. Here we strengthen this theory by providing additional results (which we had announced in [V. Barbu, I. Lasiecka, R. Triggiani, Extended algebraic Riccati equations in the abstract hyperbolic case, Non-linear Analysis 40 (2000) 105-129] and [R. Triggiani, The algebraic Riccati equation with unbounded control operator: the abstract hyperbolic case revisited, AMS, Contemporary Mathematics 209 (1997) 315-338]): in particular that PD(AF)=D(A∗) and that PD(A)=D(AF∗).