Introduction.Positive selfadjoint extensions of a symmetric operator have been investigated by many mathematicians : J. von Neumann, K. Friedrichs, M. Krein, M. Birman and others. Especially M. Krein [3] observed the class of all positive selfadjoint extensions of a given positive symmetric operator, and proved among others that, in case of a densely denned operator, the greatest and the smallest positive selfadjoint extension exist. The greatest extension is shown to coincide with the extension, established by Friedrichs, while the smallest one coincides with the extension, considered by von Neumann in case of a strongly positive operator.In this paper, starting with the well known theorem of Friedrichs, we shall investigate the structure of the greatest extension (Friedrichs extension) T μ and the smallest one (von Neumann extension) T M of a given positive symmetric operator T from various points of view. Theorem 1 gives a necessary and sufficient condition for T with non-dense domain to admit positive selfadjoint extensions. If any one of such extensions exists, the von Neumann extension is shown to exist, and its domain is explicitly determined. Among many consequences of this theorem is a simple description of the von Neumann extension, when it is bounded (Theorem 2). In contrast with the identity: (T + α) μ -a = T β for all positive number a, (!T + ά) M -a varies largely according to α. Theorem 3 shows that (T + a) M -a converges, in a natural sense, to the von Neumann extension T M or to the Friedrichs extension T μ according as α->0 or->oo. This theorem permits us to determine the spectrum of T M or T μ , when T M is compact or T μ has compact resolvent.
Preliminaries.A linear operator T on a Hubert space is, by definition, symmetric if (Tf,g) = (f,Tg) (f,gzD(T));here the domain D(T) is not assumed to be dense. T is called positive (resp. strongly positive), if (T/,/)^0 (resp. ^ £(/,/) for a constant £>0).
