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A characterization of positive self-adjoint extensions and its application to ordinary differential operators


Authors: Guangsheng Wei and Yaolin Jiang
Journal: Proc. Amer. Math. Soc. 133 (2005), 2985-2995
MSC (2000): Primary 47A20; Secondary 47E05, 34L05
DOI: https://doi.org/10.1090/S0002-9939-05-07837-8
Published electronically: March 22, 2005
MathSciNet review: 2159777
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Abstract: A new characterization of the positive self-adjoint extensions of symmetric operators, $T_0$, is presented, which is based on the Friedrichs extension of $T_0,$ a direct sum decomposition of domain of the adjoint $T_0^{*}$ and the boundary mapping of $T_0^{*}$. In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential operators of order $2n$ in terms of boundary conditions.


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  • 1. A. Alonso and B. Simon, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory 4 (1980), 251-270. MR 0595414 (81m:47038)
  • 2. Y. Arlinskii and E. Tsekanovskii,On von Neumann's problem in extension theory of nonnegative operators, Proc. Amer. Math. Soc. 131 (2003), 3143-3154. MR 1992855 (2004h:47034)
  • 3. M. S. Birman, On the self-adjoint extensions of positive definite operators, Mat. Sb. 38 (1956), 431-450. MR 0080271 (18:220d)
  • 4. W. N. Everitt and A. Zettl, Differential operators generated by a countable number of quasi-differential expressions on the real line, Proc. London Math. Soc. 64 (1992), 524-544. MR 1152996 (93k:34182)
  • 5. K. Friedrichs, Spektraltheorie halbbeschrankter operatoren, Math. Ann. 109 (1934), 465-487.
  • 6. M. G. Krein, The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, I, Mat. Sbornik 20 (1947), 431-495 (in Russian). MR 0024574 (9:515c)
  • 7. M. G. Krein, The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, II, Mat. Sbornik 21 (1947), 365-404 (in Russian). MR 0024575 (9:515d)
  • 8. M. Marletta and A. Zettl,The Friedrichs extension of singular differential operators, J. Differential Equations 160 (2000), 404-421. MR 1736997 (2000m:47058)
  • 9. M. Möller and A. Zettl, Symmetric differential operators and their Friedrichs extension, J. Differential Equations 115 (1995), 50-69. MR 1308604 (96a:34161)
  • 10. N. A. Naimark, Linear Differential Operators, vol. II, Ungar, New York, 1968. MR 0262880 (41:7485)
  • 11. H.-D. Niessen and A. Zettl,Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues, Proc. London Math. Soc. 64 (1992), 545-578. MR 1152997 (93e:47060)
  • 12. G. Wei, A new description of self-adjoint domains of symmetric operators, J. of Inner Monogolia University 27 (1996), 305-310. (in Chinese). MR 1440615 (98c:47031)
  • 13. G. Wei and J. Wu, Characterization of left-definiteness of Sturm-Liouville problems, Math. Nachr., 2004 (accepted for publication).
  • 14. G. Wei and Z. Xu, A characterization of boundary conditions for regular Sturm-Liouville problems which have the same lowest eigenvalues, Rocky Mountain J. Math. 2003 (accepted for publication).
  • 15. G. Wei and Z. Xu, On self-adjoint extensions of symmetric differential operators with countably infinite deficiency indices, Advances in Math. 29 (2000), 227-234. (in Chinese). MR 1789424 (2001i:47071)
  • 16. G. Wei, Z. Xu and J. Sun, Self-adjoint domains of products of differential expressions, J. Differential Equations 174 (2001), 75-90. MR 1844524 (2002d:47063)
  • 17. J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, Berlin/New York, 1980. MR 0566954 (81e:47001)

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Additional Information

Guangsheng Wei
Affiliation: Research Center for Applied Mathematics and Institute for Information and System Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: weimath@pub.xaonline.com, isystem@vip.sina.com

Yaolin Jiang
Affiliation: Research Center for Applied Mathematics and Institute for Information and System Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: yljiang@xjtu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-05-07837-8
Keywords: Friedrichs extension, positive self-adjoint extension, boundary condition
Received by editor(s): October 30, 2003
Received by editor(s) in revised form: May 17, 2004
Published electronically: March 22, 2005
Additional Notes: This research was supported by the National Natural Science Foundation of P. R. China (No. 10071048).
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society

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