State-space solutions to a discrete-timeℋ∞problem are given. For a given number γ>0we give a characterization of all controllers such that theℋ ∞norm of the closed-loop transfer function is less than γ The approach taken is based on the solution of two Riccati equations using the stable deflating subspace of a symplectic pencil. This ensures that no unnecessary assumptions on the ‘A’ matrix on the realization of the transfer function are needed. While the results are the direct analogues of those in the continuous time, the state-space derivation is considerably more difficult.