Factorizations of linear relations (Q1931833)

Conditions on a given pair \((A,B)\) of bounded linear operators acting on a Hilbert space which guarantee the existence of a bounded operator \(C\) such that \(A = BC\) have been well-explored in the literature. In this article, the authors consider similar problems in the context of linear relations. Let \(\mathfrak{X}, \mathfrak{Y},\mathfrak{Z}\) denote linear spaces over the real or complex field. A \textit{linear relation} (also known as a \textit{multivalued operator}) between \(\mathfrak{X}\) and \(\mathfrak{Y}\) is a linear subspace \(R\) of \(\mathfrak{X} \times \mathfrak{Y}\). Given two linear relations \(R \subseteq \mathfrak{X} \times \mathfrak{Y}\) and \(S \subseteq \mathfrak{Y} \times \mathfrak{Z}\), the product \(SR\) is the linear relation from \(\mathfrak{X}\) to \(\mathfrak{Z}\) defined by \(SR = \{ (x,z) : \text{}(x,y) \in R\) and \((y,z) \in S\) for some \(y \in \mathfrak{Y}\)