  This paper presents a state estimation approach for an uncertain linear
equation with a non-invertible operator in Hilbert space. The approach
addresses linear equations with uncertain deterministic input and noise in the
measurements, which belong to a given convex closed bounded set. A new notion
of a minimax observable subspace is introduced. By means of the presented
approach, new equations describing the dynamics of a minimax recursive
estimator for discrete-time non-causal differential-algebraic equations (DAEs)
are presented. For the case of regular DAEs it is proved that the estimator's
equation coincides with the equation describing the seminal Kalman filter. The
properties of the estimator are illustrated by a numerical example.
