Abstract
A nonnegative symmetric linear relation A 0 with defect one in a Krein space H has self-adjoint extensions which are not nonnegative. If the resolvent set of such an extension A is not empty, A has a so-called exceptional eigenvalue α. For α ≠ 0, ∞ this means that α is an eigenvalue in the open upper half-plane, or a positive eigenvalue with a nonpositive eigenvector, or a negative eigenvalue with a nonnegative eigenvector. In this paper we study these exceptional eigenvalues and their dependence on a parameter if the selfadjoint extensions of A 0 are parametrized according to M. G. Krein’s resolvent formula. An essential tool is a family of generalized Nevanlinna functions of the class N 1 and their zeros or generalized zeros of nonpositive type.
The first author was supported by the Hochschul- und Wissenschaftsprogramm des Bundes und der Länder of Germany.
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Jonas, P., Langer, H. (2007). On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space. In: Förster, KH., Jonas, P., Langer, H., Trunk, C. (eds) Operator Theory in Inner Product Spaces. Operator Theory: Advances and Applications, vol 175. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8270-4_8
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