SOME REMARKS ON KREIN EXTENSION THEORY

The extension problem of semibounded symmetric operators and symmetric operators with a pp is studied in detail. Using a suitable representation (Krein model) for the inverses of those operators a parameterization of their symmetric and self-adjoint extensions is introduced which improves Krein's famous extension theory. In particular, the parameterization clearly shows which self-adjoint extensions in the gap case correspond to Friedrichs and v. Neumann or Krein extensions in the semibounded case. Moreover, special properties of the extensions as the exactness of the gap are characterized in terms of the parameters.