Robust non-fragile feedback stabilization is studied for causal linear time-invariant (LTI) input-output systems defined in an L-2(R) signal space setup, or equivalently in an L-2(jR) setup in the frequency domain. Two-operator descriptions for the plant and the controller are used with bounded operators defined on the full space L-2(R). Basic properties of causal LTI operators on weighted L-2 spaces, L-2(R, w), which are relevant to stabilization theory, are also established for different choices of the weighting function w. Weighted L-2 spaces allow the inclusion of persistent signals such as steps and thus provide the potential of important generalizations of the robust stabilization theory on L-2( R). In particular, it is shown that there is a large class of weights w for which the L-2(R, w) setup leads to H-infinity optimization. This implies that robust H-infinity design can be applied to a large class of weighted L-2 spaces on R. Problems of closability of unstable causal LTI convolution operators are discussed. This strengthens the importance of the two-operator model Ay = Bu as the basic plant input-output model on the full time axis. It is discussed how to obtain such models with A and B bounded causal coprime operators defined on the full signal space.