Boundary relations and generalized resolvents of symmetric operators

Abstract

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in $\dC_+$) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a 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 the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.