In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic pencil (ESP). We present a general form for all linear bounded solutions of the ARS in terms of the deflating subspaces of the ESP. This relation is analogous to the results of the Hamiltonian approach for the continuous-time ARE and to the symplectic pencil approach for the finite-dimensional discrete-time ARE. In particular, we show that there is a one-to-one relation between deflating subspaces with a special structure and the solutions of the ARS.
Using the relation between the solutions of the ARS and the deflating subspaces of the ESP, we give characterizations of self-adjoint, nonnegative, and stabilizing solutions. In addition we give criteria for the discrete-time, infinite-dimensional ARE to have a maximal self-adjoint solution. Furthermore, we consider under which conditions a solution of the ARS satisfies the ARE as well.