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

TL;DR

This paper investigates least squares problems involving operators in Krein spaces, providing solvability conditions, characterizations of solutions, and a novel characterization of the Moore-Penrose inverse within this context.

Contribution

It offers a comprehensive analysis of operator least squares problems in Krein spaces, including solvability criteria and explicit solution characterizations, with a new approach to Moore-Penrose inverses.

Findings

Complete solvability conditions for the problems.

Explicit characterization of solutions including Moore-Penrose inverse.

Decomposition of operators B for addressing min-max problems.

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.