A realization theorem for generalized Nevanlinna families (Q2890635)

Boundary triplets and Weyl functions are used in the extension theory of symmetric operators and relations. By means of a boundary triplet all self adjoint extensions can be parameterized and their spectra can be described via the Weyl function. If the underlying space is a Hilbert space, then the Weyl function is a Nevanlinna function. In the case of a Pontryagin space, the Weyl function is a generalized Nevanlinna function. Conversely, every Nevanlinna or generalized Nevanlinna function with an additional strictness property can be realized as the Weyl function of a boundary triplet.NEWLINENEWLINEThe (more general) concept of boundary relations and associated Weyl families in Hilbert spaces is a generalization of the notion of a boundary triplet. It was introduced in [\textit{V. Derkach, S. Hassi, M. Malamud} and \textit{H. De Snoo}, Trans. Am. Math. Soc. 358, No. 12, 5351--5400 (2006; Zbl 1123.47004); see also Russ. J. Math. Phys. 16, No. 1, 17--60 (2009; Zbl 1182.47026)]. This generalization extends the above mentioned realization, i.e., this generalization allows to interpret all Nevanlinna functions (and Nevanlinna families) as Weyl families.NEWLINENEWLINEIt is the main feature of the present paper to repeat this kind of interpretation in a Pontryagin space setting. Therefore, the notions of boundary relations and Weyl families in a Pontryagin space are introduced. This follows the same pattern as in the Hilbert space case. Moreover, the notions of generalized Nevanlinna pairs and generalized Nevanlinna families are introduced. As the main result it is shown that every generalized Nevanlinna family can be realized as the Weyl family of a boundary relation in a Pontryagin space. Conversely, it is shown that every Weyl family corresponding to a boundary relation in a Pontryagin space is a generalized Nevanlinna family. These realizations are made in such a way that the index \(\kappa\) of indefiniteness of the Pontryagin space coincides with the number of negative squares of the corresponding generalized Nevanlinna family.