Characterizations of positive selfadjoint extensions

Sebestyén, Zoltán; Stochel, Jan

doi:10.1090/S0002-9939-06-08590-X

Characterizations of positive selfadjoint extensions
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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.