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On von Neumann's problem in extension theory of nonnegative operators


Authors: Yury Arlinskii and Eduard Tsekanovskii
Journal: Proc. Amer. Math. Soc. 131 (2003), 3143-3154
MSC (2000): Primary 47A63, 47B25; Secondary 47B65
Published electronically: February 12, 2003
MathSciNet review: 1992855
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Abstract: The solution of von Neumann's problem about parametrization of all nonegative selfadjoint extensions of a nonnegative densely defined operator in terms of his formulas is obtained.


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  • 1. Alberto Alonso and Barry Simon, The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory 4 (1980), no. 2, 251–270. MR 595414
  • 2. Tsuyoshi Ando and Katsuyoshi Nishio, Positive selfadjoint extensions of positive symmetric operators, Tôhoku Math. J. (2) 22 (1970), 65–75. MR 0264422
  • 3. Yu. M. Arlinskiĭ, Maximal sectorial extensions and closed forms associated with them, Ukraïn. Mat. Zh. 48 (1996), no. 6, 723–738 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 6, 809–827 (1997). MR 1418150, 10.1007/BF02384168
  • 4. Y. M. Arlinskii, Extremal extensions of sectorial linear relations, Mat. Stud. 7 (1997), no. 1, 81–96, 112 (English, with English and Russian summaries). MR 1683231
  • 5. Yu.Arlinskii, S.Hassi, Z.Sebestyen, H.de Snoo, On the class of extremal extensions of a nonnegative operators, Oper. Theory Adv. Appl., 127 (2001), 41-81.
  • 6. M.S.Birman, On the self-adjoint extensions of positive definite operators, Mat.Sbornik 38 (1956), 431-450 (Russian).
  • 7. Earl A. Coddington and Hendrik S. V. de Snoo, Positive selfadjoint extensions of positive symmetric subspaces, Math. Z. 159 (1978), no. 3, 203–214. MR 0500265
  • 8. V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1–95. MR 1087947, 10.1016/0022-1236(91)90024-Y
  • 9. V. A. Derkach, M. M. Malamud, and È. R. Tsekanovskiĭ, Sectorial extensions of a positive operator, and the characteristic function, Ukrain. Mat. Zh. 41 (1989), no. 2, 151–158, 286 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 2, 136–142. MR 992814, 10.1007/BF01060376
  • 10. K.Friedrichs, Spektraltheorie halbbeschrankter operatoren, Math. Ann., 109 (1934), 465-487.
  • 11. Fritz Gesztesy, Nigel J. Kalton, Konstantin A. Makarov, and Eduard Tsekanovskii, Some applications of operator-valued Herglotz functions, Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997), Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 271–321. MR 1821917
  • 12. Fritz Gesztesy and Eduard Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr. 218 (2000), 61–138. MR 1784638, 10.1002/1522-2616(200010)218:1<61::AID-MANA61>3.3.CO;2-4
  • 13. M.L.Gorbachuk, V.I.Gorbachuk, A.N.Kochubei, Extension theory for symmetric operators and boundary value problems for differential equations, Ukrainian Mat.J. 41, No.10 (1989), 1298-1313 (Russian). English translation, pp. 1117-1129.
  • 14. T.Kato, Perturbation theory for linear operators, Springer-Verlag, 1966.
  • 15. M.G.Krein, The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, I, Mat.Sbornik 20, No.3 (1947), 431-495 (Russian).
  • 16. M.G.Krein, The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, II, Mat.Sbornik 21, No.3 (1947), 365-404 (Russian).
  • 17. V.E.Lyantse, H.B.Majorga, On selfadjoint extensions of Schrödinger operator with a singular potential. Lviv university. Deposited in VINITI 15.01.81, N 240-81DEP.
  • 18. J.von Neumann, Allgemeine eigenwerttheorie Hermitescher funktionaloperatoren, Math. Ann. 102 (1929), 49-131.
  • 19. F. S. Rofe-Beketov, The numerical range of a linear relation and maximum relations, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 44 (1985), 103–112 (Russian); English transl., J. Soviet Math. 48 (1990), no. 3, 329–336. MR 800817, 10.1007/BF01101255
  • 20. M.I.Vishik, On general boundary conditions for elliptic differential equations, Trudy Moskov. Mat.Obsc., 1 (1952), 187-246 (Russian). Amer.Math.Soc.Transl. (2), 24 (1963), 107-172.

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Additional Information

Yury Arlinskii
Affiliation: Department of Mathematics, East Ukrainian National University, Kvartal Molodyozhny, 20-A, 91034, Lugansk, Ukraine
Email: yma@snu.edu.ua

Eduard Tsekanovskii
Affiliation: Department of Mathematics, P.O. Box 2044, Niagara University, New York 14109
Email: tsekanov@niagara.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06859-X
Received by editor(s): August 6, 2001
Received by editor(s) in revised form: May 6, 2002
Published electronically: February 12, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society