Representation theory of topological selections of multivalued linear mappings with applications to integral and differential operators (Q582557)

Let X, Y be real or complex Banach spaces and let a subspace \(M\subset X\times Y\) be a graph of a multivalued linear mapping (linear relation). R is an algebraic selection of M if \(R=\{a\in M:\) \(P(a)=0\}\) for an algebraic projector P on M with Range P\(=\{0\}\times M(0)\). If P is continuous then R is a topological selection. There are considered closed linear relations M with finite dimensional M(0) and \(M^+(0)\), where \(M^+\subset X^{\#}\times Y^{\#}\), \(X^{\#}\) is the dual of X, in order to generalize some results of \textit{E. A. Coddington} and \textit{A. Dijksma} [cf. Ann. Mat. Pura Appl., IV. Ser. 118, 1-118 (1978; Zbl 0408.47035)]. Let \(M_ 1\) be a closed subspace of \(X\times Y\) and let M be a closed subspace of \(M_ 1\) such that dim \(M_ 1/M<+\infty\). Conditions for some graph of a linear operator to be a topological selection of \(M^{- 1}\) and a characterization of all topological selections of \(M^{-1}\) which are compact (or integral operators in the sense of Kantorovich and Akilov), provided that there exists one such a selection, are given. One can consider some of these results as results on finite dimensional perturbation of closed linear relations. Applications to differential operators (in particular, to operators with two-points boundary conditions) are shown.