In the two block Hinftyoptimization problem, usually we are given the state-space realizations of the proper rational matrices R_{1}(s) and R_{2}(s) whose poles are all the open right-half plane. Two problems are studied in the note. The first is the evaluation of \phi(s)R_{1}(s) at s = s_{k}, k = 1, 2, ..., n , where \phi(s) is an inner function whose zeros {s_{k}, k = 1, 2, ..., n } are the poles of R_{1}(s) . This evaluation is essential if Chang and Pearson's method is used for computing the optimal Hinftynorm. The problem is solved in state space via the solutions of Lyapunov equations. Neither polynomial matrix manipulations nor numerical pole-zero cancellations are involved in the evaluation. The second problem is to find a stable state-space realization of S(s) = U(s)R_{2}(s) where U(s) is an inner matrix. This problem arises in the spectral factorization of \gamma^{2} - R_{2}^{\ast}R_{2} . Doyle and Chu had a method for constructing stable S(s) based on a minimal realization of R_{2}(s) . An alternate method is proposed. The alternate method does not require a minimal realization of R_{2}(s) and only a Lyapunov equation is involved.