The Spectra of Selfadjoint Extensions of Entire Operators with Deficiency Indices (1,1)

Abstract

We give necessary and sufficient conditions for real sequences to be the spectra of selfadjoint extensions of an entire operator whose domain may be non-dense. For this spectral characterization we use de Branges space techniques and a generalization of Krein's functional model for simple, regular, closed, symmetric operators with deficiency indices (1,1). This is an extension of our previous work in which similar results were obtained for densely defined operators. Mathematics Subject Classification(2000): 46E22, 47A25, 47B25

The class ofn-entire operators

“…Moreover, for any n ∈ Z + , one has necessary and sufficient conditions for the existence of a real zerofree entire functions in the space assoc n B. The statement of this important result (see theorem 2.5 below) is essentially theorem 3.2 of [40] with a slight modification justified by lemmas 3.3 and 3.4 of [30]. See also [39] for a more elementary (and restricted) version of this theorem.…”

“…It is worth mentioning that there is an alternative functional model for the same class S(H) recently developed in [26]. Some of the material in this subsection can also be found in [31]. The functional model described below rests on the properties of the generalized Cayley transform given in Proposition 2.2 with the following addition.…”

“…In spite of this, we will stick to (31) in order to simplify the ongoing discussion. Let us henceforth denote the spaces assoc n B(e) with the inner product (31) as B −n .…”

“…It is a bit less apparent that an operator entire in the generalized sense is indeed 1-entire. To see this we observe that, as a direct consequence of [30,Proposition 5.1], given µ ∈ H −1 \ H one can find µ 0 , µ 1 ∈ H such that [ξ(z), µ] = ξ(z), µ 0 + z ξ(z), µ 1 for all z ∈ C, hence reducing Krein's to our definition with n = 1. It is worth remarking here that the class S(H) includes operators with non-dense domain.…”

de Branges Spaces and Krein’s Theory of Entire Operators

“…This functional model, which was developed in [ST10,ST13a,ST13b], stems from Krein's theory of representation of symmetric operators developed in his original work [Kre44b, Theorems 2 and 3] (cf. [GG97, Section 1.2]), but differs from it in a crucial way as will be explained below (see Remark 4.7).…”