This paper deals with a general steady-state estimation problem in the H(infinity) setting. The existence of the stabilizing solution of the related algebraic Riccati equation ( ARE) and of the solution of the associated J-spectral factorization problem is investigated. The existence of such solutions is well established if the prescribed attenuation level gamma is larger than gamma(f) ( the infimum of the values of gamma for which a causal estimator with attenuation level gamma exists). We consider the case when gamma <= gamma(f) and show that the stabilizing solution of the ARE still exists ( except for a finite number of values of gamma) as long as a fixed-lag acausal estimator ( smoother) does. The stabilizing solution of the ARE may be employed to derive a state-space realization of a minimum-phase J-spectral factor of the J-spectrum associated with the estimation problem. This J-spectral factor may be used, in turn, to compute the minimum-lag smoothing estimator. Some of the aspects of the J-spectral factorization problem and the properties of its solutions are discussed in correspondence to the (finite number of) values of gamma for which the stabilizing solution of the ARE does not exist.