Boundary relations and generalized resolvents of symmetric operators

By V. Derkach, S. Hassi, M. Malamud, H. D. de Snoo. 2006

114 citations recorded by the source corpus.

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/97d427553bbd8fec2c59cce16130aa02ddd39610

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, background): “Other generalizations of boundary triples and abstract concepts of boundary mappings were studied, e.g. in [Ar- linskĭı, 2000; Derkach et al., 2006, 2009; Kopachevskĭı and Krĕın, 2004; Posilicano, 2004, 2008; Mogilevskĭı, 2006, 2009], The aim of the present paper is to give an introduction to…”