We study the existence of general finite-dimensional compensators in connection with the H-infinity-optimal control of linear time-invariant systems on a Hilbert space with noisy output feedback. The approach adopted uses a Galerkin-type approximation, where there is no requirement for the system operator to have a complete set of eigenvectors. We show that if there exists an infinite-dimensional compensator delivering a specific level of attenuation, then a finite-dimensional compensator exists and achieves the same level of disturbance attenuation. In this connection, we provide a complete analysis of the approximation of infinite-dimensional generalized Riccati equations by a sequence of finite-dimensional Riccati equations. As an illustration of the theory developed here, we provide a general procedure for constructing finite-dimensional compensators for robust control of flexible structures.