A characterization

of semibounded selfadjoint operators

Authors:
Seppo Hassi, Michael Kaltenbäck and Henk de Snoo

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2681-2692

MSC (1991):
Primary 47B15, 47B25

DOI:
https://doi.org/10.1090/S0002-9939-97-03960-9

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a class of closed symmetric operators with defect numbers it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator is semibounded if and only if each one-dimensional restriction of has a generalized Friedrichs extension.

**[1]**E.A. Coddington and H.S.V. de Snoo,*Positive selfadjoint extensions of positive symmetric subspaces*, Math. Z.,**159**(1978), 203-214. MR**58:17936****[2]**S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo,*Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass of Nevanlinna functions*, J. Operator Theory, to appear.**[3]**S. Hassi, H. Langer, and H.S.V. de Snoo,*Selfadjoint extensions for a class of symmetric operators with defect numbers*, 15th OT Conference Proc., (1995), 115-145.**[4]**S. Hassi and H.S.V. de Snoo,*One-dimensional graph perturbations of selfadjoint relations*, Ann. Acad. Sci. Fenn., Series A.I. Math.,**22**(1997), 123-164.**[5]**I.S. Kac and M.G. Kre[??]in,*-functions-analytic functions mapping the upper halfplane into itself*, Supplement I to the Russian edition of F.V. Atkinson,*Discrete and continuous boundary problems*, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2)**103**(1974), 1-18). MR**48:6969****[6]**T.Kato,*Perturbation theory for linear operators*, Springer-Verlag, Berlin-Heidelberg-New York, 1966. MR**34:3324****[7]**A.G.R. McIntosh,*Hermitian bilinear forms which are not semibounded*, Bull. Amer. Math. Soc.,**76**(1970), 732-737. MR**41:5988**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47B15,
47B25

Retrieve articles in all journals with MSC (1991): 47B15, 47B25

Additional Information

**Seppo Hassi**

Affiliation:
Department of Statistics University of Helsinki PL 54, 00014 Helsinki Finland

Email:
hassi@cc.helsinki.fi

**Michael Kaltenbäck**

Affiliation:
Institut für Analysis, Technische Mathematik und Versicherungsmathematik Technische Universität Wien Wiedner Hauptstrasse 8-10/114 A-1040 Wien Österreich

Email:
mbaeck@geometrie.tuwien.ac.at

**Henk de Snoo**

Affiliation:
Department of Mathematics University of Groningen Postbus 800, 9700 AV Groningen Nederland

Email:
desnoo@math.rug.nl

DOI:
https://doi.org/10.1090/S0002-9939-97-03960-9

Keywords:
Symmetric operator,
selfadjoint extension,
Friedrichs extension,
$Q$-function,
Nevanlinna function

Received by editor(s):
April 22, 1996

Additional Notes:
The second author was supported by “Fonds zur Förderung der wissenschaftlichen Forschung” of Austria, Project P 09832-MAT

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society