Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces

Abstract

Given a Krein space H and B, C in L(H), L(H), the bounded linear operators on H, the minimization/maximization of expressions of the form (BX−C)#(BX−C) as X runs over L(H) is studied. Complete solutions are found for the problems posed, including solvability criteria and a characterization of the solutions when they exist. Min-max problems associated to Krein space decompositions of B are also considered, leading to a characterization of the Moore-Penrose inverse as the unique solution of a variational problem. Other generalized inverses are similarly described.