A criterion for selfadjointness in Krein spaces

Abstract

The main outcome of this paper is a tool for investigating selfadjointness of symmetric operators in a Krein space. Our result covers certain previous theorems in domination theory in a Hilbert and Krein space. It is also applied to obtain a sufficient condition for selfadjointness of a first-order differential operator.

On determining the domain of the adjoint operator

“…Defining A + as the adjoint of A with respect to [·, −] we easily see that A + = H −1 A * H. Hence, Theorem 1 can suite as a criterion for selfadjointness of a closed symmetric operator in a Krein space, cf. [26]. Nevertheless, we will not use the indefinite inner product and the operator A + in the present paper.…”

“…Remark 5. It was shown in [26] that in the (a4) case conditions (d1) the operators T A and AT are bounded and the domain of the commutator D(ad(T, A)) is dense in K; (d2) the operator A 0 T * is densely defined.…”

“…The method presented above is an alternative approach, based rather on the notions of commutativity and domination. An example of a first order symmetric differential operator from [26] shows the difference between those two approaches.…”

“…Section 1 has a preliminary character, but already in the consecutive section we prove the main result of the paper. In Section 3 we consider the class (a1) of formally normal operators, extending the results from [26]. In Section 4 we will consider H-symmetric operators (class (a4)) given by infinite matrices.…”

The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

“…In particular, the Schrödinger operators could be unbounded from below and from above, see [Gol] for instance. Unlike in [Ber], we rely on an commutator approach, see [Wo1,Wo2] for similar techniques. The points (3) and (4) follow by application of the Nelson commutator Theorem.…”