Weighted projections into closed subspaces

TL;DR

This paper explores $A$-projections in Hilbert spaces, relating them to compatibility of positive operators and closed subspaces, and examines their connection to weighted least squares problems.

Contribution

It extends the concept of $A$-projections to infinite-dimensional spaces and clarifies their relationship with compatibility and weighted least squares.

Findings

Characterization of $A$-projections in Hilbert spaces

Connection between $A$-projections and compatibility of operators

Insights into weighted least squares problems

Abstract

In this paper we study $A$-projections, i.e. operators of a Hilbert space $\HH$ which act as projections when a seminorm is considered in $\HH$. $A$-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite dimensional spaces. We relate this concept to the theory of compatibility between positive operators and closed subspaces of $\HH$. We also study the relationship between weighted least squares problems and compatibility.