Self-adjoint Extensions: Cayley Transform and Krein Transform

Abstract

In Chap. 13, we treat two fundamental methods of self-adjoint extensions. The first one, due to J. von Neumann, uses the Cayley transform and reduces the self-adjoint extension problem for a densely defined symmetric operator to the problem of unitary extensions of its Cayley transform. The second one, due to M.G. Krein, is based on the Krein transform and describes positive self-adjoint extensions of a densely defined positive symmetric operator by means of bounded self-adjoint extensions of its Krein transform. A classical theorem of T. Ando and K. Nishio characterizes when a (not necessarily densely defined) positive symmetric operator has a positive self-adjoint extension and shows that then a smallest such extension, called the Krein–von Neumann extension, exists. We prove this result and derive a number of interesting applications. In the final section, we deal with two special situations for the construction of self-adjoint extensions. These are symmetric operators commuting with a conjugation or anticommuting with a symmetry.