Boundary Triples, Weyl Functions, and the Krein Formula

This chapter provides an overview of recent developments of Calkin’s approach to extension theory of symmetric operators. We recall the notions of ordinary and unitary boundary triples and the corresponding Weyl functions of symmetric operators in Hilbert and Krein spaces and present descriptions of different classes of extensions of such operators in terms of boundary triples. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triples approach. In particular, descriptions of generalized resolvents of nonnegative operators and operators with finite number of negative squares are presented. M.G. Krein’s representation theory for symmetric operators in Hilbert and Krein spaces and the theory of \({\mathfrak L}\) -resolvents of different classes of symmetric operators in connection with the theory of boundary triples are discussed. Applications to different continuation problems related to the extension theory of symmetric operators in Hilbert and Pontryagin spaces are given.