Operator extensions with closed range

Abstract

We characterize those positive (not necessarily densely defined) operators whose Krein-von Neumann extension, the smallest among all positive selfadjoint extensions, has closed range. In addition, we construct their Moore- Penrose pseudoinverse by employing factorization via an auxiliary Hilbert space. Other extremal extensions, in particular the Friedrichs extension, are also in- vestigated from this point of view. As an application, new characterizations of essentially selfadjoint positive operators are presented.