Abstract. For a class of closed symmetric, not necessarily semibounded, operators with defect numbers $(1,1)$, we introduce a generalization of the Kreĭn-von Neumann extension. By a formal inversion of the graphs of the underlying linear operators or relations this extension is related to a generalization of the Friedrichs extension. We characterize the Kreĭn--von Neumann extension in different ways, by the so-called $Q$-functions and by certain quadratic forms. Models involving triplet spaces or the spectral measure are also presented.

AMS Subject Classification (1991): 47A70, 47B15, 47B25, 47A55, 47A57

Keyword(s): Symmetric operator, selfadjoint extension, Friedrichs extension, Kreĭn-von Neumann extension, Q-function, Nevanlinna function, operator model, domain perturbation

Received October 28, 1997 and in revised form July 31, 1998. (Registered under 3313/2009.)