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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal extensions for the sum of nonnegative selfadjoint relations


Authors: Seppo Hassi, Adrian Sandovici, Henk de Snoo and Henrik Winkler
Journal: Proc. Amer. Math. Soc. 135 (2007), 3193-3204
MSC (2000): Primary 47A57, 47B25; Secondary 47A55, 47B65
Published electronically: May 14, 2007
MathSciNet review: 2322750
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Abstract: The sum $ A+B$ of two nonnegative selfadjoint relations (multi-valued operators) $ A$ and $ B$ is a nonnegative relation. The class of all extremal extensions of the sum $ A+B$ is characterized as products of relations via an auxiliary Hilbert space associated with $ A$ and $ B$. The so-called form sum extension of $ A+B$ is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with $ A$ and $ B$. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on $ A$ and $ B$.


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Additional Information

Seppo Hassi
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland
Email: sha@uwasa.fi

Adrian Sandovici
Affiliation: Colegiul Naţional “Petru Rareş”, 610101, Str. Ştefan cel Mare, Nr. 4, Piatra Neamt, Romania
Email: adrian.sandovici@yahoo.com

Henk de Snoo
Affiliation: Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, Nederland
Email: desnoo@math.rug.nl

Henrik Winkler
Affiliation: Institut für Mathematik, MA 6-4, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Deutschland
Email: winkler@math.tu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08827-2
PII: S 0002-9939(07)08827-2
Keywords: Nonnegative selfadjoint relation, Friedrichs extension, Kre\u{\i}n-von Neumann extension, extremal extension, form sum extension
Received by editor(s): March 27, 2006
Received by editor(s) in revised form: June 15, 2006
Published electronically: May 14, 2007
Additional Notes: The fourth author was supported by the “Fond zur Förderung der wissenschaftlichen Forschung” (FWF, Austria), grant number P15540-N05.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.