Quotient morphisms, compositions, and Fredholm index (Q734917)

Let \({\mathcal X}\) be a vector space, let \(\text{Lat}({\mathcal X})\) be the lattice consisting of all vector subspaces of \({\mathcal X}\), and let \(Q({\mathcal X})\) be the family of all quotient spaces of the form \(X/X_0\) with \(X_0,X\in\text{Lat}({\mathcal X})\), \(X_0\subset X\). A quotient morphism from \({\mathcal X}\) to \({\mathcal Y}\) is any linear map \(T:X/X_0\to Y/Y_0\), where \(X/X_0\in Q({\mathcal X})\) and \(Y/Y_0\in Q({\mathcal Y})\). The authors define a composition between quotient morphisms and prove a multiplication formula for the index of the composition of two Fredholm quotient morphisms. These results are applied to linear relations, which are subspaces \(Z\in\text{Lat}({\mathcal X}\times {\mathcal Y})\) for given linear spaces \({\mathcal X}\) and \({\mathcal Y}\). Thus, they obtain a multiplication formula for the index of the composition of two Fredholm linear relations. Note that a version of the latter result has been recently obtained in \textit{A.\,Sandovici} and \textit{H.\,de Snoo} [``An index formula for the product of linear relations'', Linear Algebra Appl.\ 431, No.\,11, 2160--2171 (2009; Zbl 1176.47014)].