Steepest descent for singular linear operators with nonclosed range

Abstract

Let T be a bounded linear operator between two Hilbert spaces with the range of T not necessarily closed, and let T ^† denote the generalized inverse of T. The method of steepest descent for minimizing is shown to converge monotonically starting with x ₀ = 0, to T ^† y for any y whose orthogonal projection on the closure of the range of T, is in the range of TT ^*. This set of y's is dense in the domain of T†. The method is also applied to generalized least squares solutions of a class of unbounded linear operator equations