  An algebraic Riccati equation for linear operators is studied, which arises
in systems theory. For the case that all involved operators are unbounded, the
existence of infinitely many selfadjoint solutions is shown. To this end,
invariant graph subspaces of the associated Hamiltonian operator matrix are
constructed by means of a Riesz basis with parentheses of generalised
eigenvectors and two indefinite inner products. Under additional assumptions,
the existence and a representation of all bounded solutions is obtained. The
theory is applied to Riccati equations of differential operators.
