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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
DOI: https://doi.org/10.1090/S0002-9939-07-08827-2
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$.


References [Enhancements On Off] (What's this?)

  • 1. Yu.M. Arlinskii, S. Hassi, Z. Sebestyén, and H.S.V. de Snoo, ``On the class of extremal extensions of a nonnegative operator'', Oper. Theory: Adv. Appl. (B. Sz.-Nagy memorial volume), 127 (2001), 41-81. MR 1902794 (2003d:47028)
  • 2. Yu.M. Arlinskii and E.R. Tsekanovskii, ``Quasi selfadjoint contractive extensions of Hermitian contractions'', Teor. Funkts., Funkts. Anal. Prilozhen, 50 (1988), 9-16.MR 0975668 (90b:47014)
  • 3. E.A. Coddington, Extension theory of formally normal and symmetric subspaces, Mem. Amer. Math. Soc., 134, 1973. MR 0477855 (57:17357)
  • 4. B. Farkas and M. Matolcsi, ``Commutation properties of the form sum of positive, symmetric operators'', Acta Sci. Math. (Szeged), 6 (2001), 777-790.MR 1876466 (2003a:47050)
  • 5. B. Farkas and M. Matolcsi, ``Positive forms on Banach spaces'', Acta Math. Hungar., 99 (2003), 43-55.MR 1973084 (2004d:47004)
  • 6. P.A. Fillmore and J.P. Williams, ``On operator ranges'', Adv. Math., 7 (1971), 254-281.MR 0293441 (45:2518)
  • 7. S. Hassi, M. M. Malamud, and H. S. V. de Snoo, ``On Krein's extension theory of nonnegative operators'', Math. Nachr., 274/275 (2004), 40-73.
  • 8. S. Hassi, A. Sandovici, H.S.V. de Snoo, and H. Winkler, ``Form sums of nonnegative selfadjoint operators'', Acta Math. Hungar., 111 (2006), 81-105.MR 2188974
  • 9. S. Hassi, A. Sandovici, H.S.V. de Snoo, and H. Winkler, ``A general factorization approach to the extension theory of nonnegative operators and relations'', J. Operator Theory, to appear.
  • 10. Z. Sebestyén and J. Stochel, ``Restrictions of positive self-adjoint operators'', Acta Sci. Math. (Szeged), 55 (1991), 149-154.MR 1124953 (92i:47024)
  • 11. Z. Sebestyén and J. Stochel, ``On products of unbounded operators'', Acta Math. Hungar., 100 (1-2) (2003), 105-129.MR 1984863 (2004c:47003)

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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: https://doi.org/10.1090/S0002-9939-07-08827-2
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.

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