Journal of Operator Theory

Volume 37, Issue 1, Winter 1997 pp. 155-181.

Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass

$\mathbbN_1$ of Nevanlinna functions

Authors: Seppo Hassi

$1$ , Michael KaltenbÃ¤ck

$2$ and Henk S.V. de Snoo

$3$
Author institution:

$1$ Department of Statistics, University of Helsinki, PL 54, 00014 Helsinki, FINLAND

$2$ Institut fÃ¼r Analysis, Technische Mathematik und Versicherungsmathematik, Technische UniversitÃ¤t Wien, Wiedner Hauptstrasse 8-10/114, A-1040 Wien, Ã–STERREICH

$3$ Department of Mathematics, University of Groningen, Postbus 800, 9700 AV Groningen, NEDERLAND

Summary: The selfadjoint extensions of a closed symmetric operator S with defect numbers

$1, 1$ are described when S has a Q-function belonging to the subclass N_1 of all Nevanlinna functions. With the associated triplet of Hilbert spaces

$\mathfrak{H}_{ + 1} \subset \mathfrak{H} \subset \mathfrak{H}_{ - 1}$ all but one of the selfadjoint extensions of S are interpreted as rank one perturbations of a fixed operator extension; the exceptional extension corresponds to a proper relation extension. Each nonexceptional selfadjoint extension gives rise to the same triplet of Hilbert spaces. The exceptional extension is characterized in a similar way as the Friedrichs extension of a semibounded operator.

Keywords: Symmetric operator, selfadjoint extension, rank one perturbation, Friedrichs extension, Q-function, Nevanlinna function, triplet of Hilbert spaces.

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