# Self-adjoint Extensions: Cayley Transform and Krein Transform

Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)

## 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.

## Keywords

Cayley Transform Positive Self-adjoint Extension Positive Symmetric Operator Isometric Operator Partial Isometry
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Classical Articles

1. [Kr3]
Krein, M.G.: The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. Mat. Sb. 20, 365–404 (1947)
2. [vN2]
Von Neumann, J.: Zur Theorie der unbeschränkten Matrizen. J. Reine Angew. Math. 161, 208–236 (1929)

## Articles

1. [ALM]
Ando, T., Li, C.-K., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)
2. [Dr]
Dritschel, M.: On factorization of trigonometric polynomials. Integral Equ. Oper. Theory 49, 11–42 (2004)
3. [Sch2]
Schmüdgen, K.: On domains of powers of closed symmetric operators. J. Oper. Theory 9, 53–75 (1983)
4. [Sch4]
Schmüdgen, K.: A formally normal operator having no normal extension. Proc. Am. Math. Soc. 98, 503–504 (1985) Google Scholar
5. [WN]
Woronowicz, S.L., Napiorkowski, K.: Operator theory in the C -algebra framework. Rep. Math. Phys. 31, 353–371 (1992)