On the solvability of a system of forward-backward linear equations with unbounded operator coefficients

Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

“…We prove that there exists a unique solution of the Riccati integral equation for strongly continuous operator functions ranging in the space L(X, X * ), where X is an arbitrary reflexive Banach space. It is important to note that, in contrast to the papers [8,5], we do not assume an embedding between the space X and the dual space.…”

“…Let P ∈ C s (I; L(X 1 , X 2 )), let a forward evolution famuily ← − Ψ t,s in L(X 1 ) be a solution of Eq. (8), and let a backward evolution family − → Ψ s,t in L(X 2 ) be a solution of the equation…”

“…By ← − Ψ If P * n (t) = P n (t) for all t ∈ I, then it follows from Eqs. (8) and (10), the selfadjointness of the operator function B(t) and the condition − →…”

“…A triple X ֒→ H ֒→ X * of spaces with dense embeddings was considered in [5] for a Hilbert space X and in [8] for a reflexive Banach space X. In these papers, the solvability of an autonomous Riccati equation in operator functions ranging in the spaces L(X * , X) and L(X, X * ), respectively, was established.…”

“…(11) in which the evolution families ← − Ψ r,t and − → Ψ t,r are determined by Eqs. (8) and (10), respectively. It follows from Proposition 6 that the operator function P ∈ C s (I; L(X, X * )) is a solution of the Riccati integral equation (1).…”