Classical Self-Adjoint Extension Schemes

Abstract

This chapter presents the two classical self-adjoint extension theories a la von Neumann and a la Kreı̆n, Višik, and Birman. We start with the discussion of the Friedrichs extension, which is in fact an independent quadratic form construction, and after that we present the theory of the Cayley transform for symmetric operators, on which von Neumann’s extension scheme was originally based and is indeed presented here. We also supplement this with the reinterpretation made by N. Dunford and J. T. Schwartz, where the emphasis is rather put on the abstract boundary value problem. Next, we discuss the Kreı̆n transform of positive operators and its role in Kreı̆n’s theory of self-adjoint extensions, that parallels the role of the Cayley transform in von Neumann’s extension theory. We then present in its entirety Kreı̆n’s extension theory of symmetric semi-bounded operators, following the original work by Kreı̆n, and also discussing the Ando-Nishio variant for the characterisation of the Kreı̆n-von Neumann extension. After this, the Višik-Birman parametrisation of self-adjoint extensions is analysed in its original form and in its more typical modern re-parametrisation, and all its fundamental ancillary results are discussed on invertibility of the extensions, their semi-boundedness, their spectrum, and Kreı̆n formula type resolvent identities. A detour on that sub-class of extensions that retain the same Friedrichs lower bound concludes such a review.