Characterizations of positive selfadjoint extensions
HTML articles powered by AMS MathViewer
- by Zoltán Sebestyén and Jan Stochel PDF
- Proc. Amer. Math. Soc. 135 (2007), 1389-1397 Request permission
Abstract:
The set of all positive selfadjoint extensions of a positive operator $T$ (which is not assumed to be densely defined) is described with the help of the partial order which is relevant to the theory of quadratic forms. This enables us to improve and extend a result of M. G. Krein to the case of not necessarily densely defined operators $T$.References
- Tsuyoshi Ando and Katsuyoshi Nishio, Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J. (2) 22 (1970), 65–75. MR 264422, DOI 10.2748/tmj/1178242861
- Yu. M. Arlinskiĭ, S. Hassi, Z. Sebestyén, and H. S. V. de Snoo, On the class of extremal extensions of a nonnegative operator, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 41–81. MR 1902794
- Yury Arlinskiĭ and Eduard Tsekanovskiĭ, On von Neumann’s problem in extension theory of nonnegative operators, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3143–3154. MR 1992855, DOI 10.1090/S0002-9939-03-06859-X
- Yury Arlinskiĭ and Eduard Tsekanovskiĭ, The von Neumann problem for nonnegative symmetric operators, Integral Equations Operator Theory 51 (2005), no. 3, 319–356. MR 2126815, DOI 10.1007/s00020-003-1260-x
- T. Ja. Azizov, I. S. Iohvidov, and V. A. Štraus, The normally solvable extensions of a closed symmetric operator, Mat. Issled. 7 (1972), no. 1(23), 16–26 (Russian). MR 0308842
- M. Š. Birman, On the theory of self-adjoint extensions of positive definite operators, Mat. Sb. N.S. 38(80) (1956), 431–450 (Russian). MR 0080271
- M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782, DOI 10.1007/978-94-009-4586-9
- H. Freudenthal, Über die Friedrichssche Fortsetzung halbbeschränkter Operatoren, Akademie van Wetenschappente Amsterdam, Proceedings, ser. A, 39 (1936), 832-833.
- Kurt Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1934), no. 1, 465–487 (German). MR 1512905, DOI 10.1007/BF01449150
- Jan Janas and Jan Stochel, Selfadjoint operator matrices with finite rows, Ann. Polon. Math. 66 (1997), 155–172. Volume dedicated to the memory of Włodzimierz Mlak. MR 1438336, DOI 10.4064/ap-66-1-155-172
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
- M. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 431–495 (Russian, with English summary). MR 0024574
- Vilmos Prokaj and Zoltán Sebestyén, On extremal positive operator extensions, Acta Sci. Math. (Szeged) 62 (1996), no. 3-4, 485–491. MR 1430868
- Vilmos Prokaj and Zoltán Sebestyén, On Friedrichs extensions of operators, Acta Sci. Math. (Szeged) 62 (1996), no. 1-2, 243–246. MR 1412931
- Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris; Akadémiai Kiadó, Budapest, 1965 (French). Quatrième édition. Académie des Sciences de Hongrie. MR 0179567
- Zoltán Sebestyén, On ranges of adjoint operators in Hilbert space, Acta Sci. Math. (Szeged) 46 (1983), no. 1-4, 295–298. MR 739046
- Zoltán Sebestyén, Positivity of operator products, Acta Sci. Math. (Szeged) 66 (2000), no. 1-2, 287–294. MR 1768867
- Zoltán Sebestyén and Eszter Sikolya, On Krein-von Neumann and Friedrichs extensions, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 323–336. MR 1991670
- Zoltán Sebestyén and Jan Stochel, Restrictions of positive selfadjoint operators, Acta Sci. Math. (Szeged) 55 (1991), no. 1-2, 149–154. MR 1124953
- Z. Sebestyén and J. Stochel, On products of unbounded operators, Acta Math. Hungar. 100 (2003), no. 1-2, 105–129. MR 1984863, DOI 10.1023/A:1024660318703
- Zoltán Sebestyén and Jan Stochel, Reflection symmetry and symmetrizability of Hilbert space operators, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1727–1731. MR 2120258, DOI 10.1090/S0002-9939-04-07705-6
- Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877, DOI 10.1090/coll/015
- Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954, DOI 10.1007/978-1-4612-6027-1
- J. von Neumann, Allgemeine Eigenwerttheorie Hermitischer Funktionaloperatoren, Math. Annalen 102 (1929), 49-131.
Additional Information
- Zoltán Sebestyén
- Affiliation: Department of Applied Analysis, Eötvös L. University, Pázmány Péter sétány 1/c., Budapest H-1117, Hungary
- Email: sebesty@cs.elte.hu
- Jan Stochel
- Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30059 Kraków, Poland
- Email: stochel@im.uj.edu.pl
- Received by editor(s): June 30, 2005
- Received by editor(s) in revised form: November 29, 2005
- Published electronically: October 27, 2006
- Additional Notes: The research of the second author was supported by KBN grant 2 P03A 037 024
- Communicated by: Joseph A. Ball
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1389-1397
- MSC (2000): Primary 47A20, 47B25; Secondary 47A63
- DOI: https://doi.org/10.1090/S0002-9939-06-08590-X
- MathSciNet review: 2276647
Dedicated: Dedicated to Henk de Snoo on the occasion of his sixtieth birthday.