Non-semi-bounded closed symmetric forms associated with a generalized Friedrichs extension

The theory of closed sesquilinear forms in the non-semi-bounded situation exhibits some new features, as opposed to the semi-bounded situation. In particular, there can be more than one closed form associated with the generalized Friedrichs extension SF of a non-semi-bounded symmetric operator S (if SF exists). However, there is one unique form [·, ·] satisfying Kato's second representation theorem and, in particular, dom = dom ∣SF∣1/2. In the present paper, another closed form [·, ·], also uniquely associated with SF, is constructed. The relation between these two forms is analysed and it is shown that these two non-semi-bounded forms can indeed differ from each other. Some general criteria for their equality are established. The results induce solutions to some open problems concerning generalized Friedrichs extensions and complete some earlier results about them in the literature. The study is connected to the spectral functions of definitizable operators in Kreĭn spaces.

Definitizable Operators on a Krein Space

Let A be a bounded linear operator on a Hilbert space H. Assume that A is selfadjoint in the indefinite inner product defined by a selfadjoint, bounded, invertible linear operator G on H; [x,y] := (Gx,y). In the first part of the paper we define two orders of neutrality for the pair (G, A) and a connection is made with the "types" of numbers in the point and approximate point spectrum of A. The main results of the paper are in the second part and they deal with strong and uniform definitizability of a bounded selfadjoint operator on a Pontrjagin space. They state:A) Let A be a bounded strongly definitizable operator on a Pontrjagin space ΠK, then A is uniformly definitizable.B) A bounded selfadjoint operator A on a Pontrjagin space ΠK is uniformly definitizable if and only if all the eigenvalues of A are of definite type and all the nonisolated eigenvalues of A are of positive type.Some applications to the theory of linear selfadjoint operator pencils are given.