Positive linear relations between Riesz spaces (Q346420)

Let \(E\) and \(F\) be vector lattices and let \(G(T) \subseteq E \times F\) be a vector subspace. We may associate a multi-valued linear operator \(T\) to \(G(T)\) and consider \(G(T)\) as the graph of \(T\). Such a multi-valued operator \(T\) is often called a \textit{linear relation}. In case that \(T\) is a single-valued and everywhere defined operator, one calls \(T\) \textit{positive} if \(Tx \geq 0\) for all \(x \geq 0\). Such positive operators have been, and are still, intensively studied. The authors show how the notion of positivity can be generalised to the case where \(T\) is a multi-valued operator and not necessarily everywhere defined. Then they study the structure of such positive linear relations. In particular, they consider the question whether the sum and the product of two positive relations is again positive and they prove a result about automatic boundedness of positive relations. They also demonstrate how the definition of a \textit{lattice homomorphism} can be adapted to the multi-valued setting. The paper concludes with a brief section about the eigenvalues of positive relations.