Operator least squares problems and Moore-Penrose inverses in Krein spaces

Abstract

A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX - C)^{#}(BX -C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a complete answer to the problem, including solvability conditions and characterization of the solutions, is presented. Also, an adequate decomposition of B is considered and the min-max problem is addressed. As a by-product the Moore-Penrose inverse of B is characterized as the only solution of a variational problem. Other generalized inverses are described in a similar fashion as well.