Positive selfadjoint extensions of positive symmetric operators

Abstract

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).

The Proper Dissipative Extensions of a Dual Pair

“…We draw the reader's attention to a slight difference in the way, Proposition 4.3 was stated in [3], where the supremum is taken over all f ∈ D(V ) (without the extra condition that V f = 0), which only makes sense if one assumes that ker V = {0}. The extra condition V f = 0 is a remedy for this problem and is a direct result from the reasoning of [3].…”

“…It will be the square root of the selfadjoint Kreȋn-von Neumann extension of V -denoted by V 1/2 K -which will play an important role for the results obtained in Section 4. The description of V 1/2 K obtained by Ando and Nishio [3] will allow us to give a necessary and sufficient condition (Theorem 4.7) for an extension of (A, A) to be dissipative, which we only have to check on the space by which we extend the operator A rather than on the whole domain of the extension. From this result, we proceed to give a description of all dissipative extensions of the dual pair (A, A) in terms of contractions from one "small" auxiliary space to another.…”

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…We'll be more expansive than absolutely necessary, in part, because we'll need this when we briefly turn to N -body scattering in Sects. [13][14][15] and, in part, because the elegant formalism, which I learned from Sigalov-Sigal [566] (see also Hunziker-Sigal [264]), deserves to be better known.…”

Characterizations of positive selfadjoint extensions

“…In the present paper we continue our investigations of the extension problem we started in [19] (see [1] for the case of closed positive operators and [16,17,21] for the case of bounded positive operators; see also [12,6,5,13,14,2,20,18,3,4] for related papers). Our aim is to describe the set of all positive selfadjoint extensions of a given positive operator (not necessarily densely defined) using the language of If A is a densely defined operator in H, then its adjoint is denoted by A * .…”