About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 676835, 16 pages
http://dx.doi.org/10.5402/2012/676835
Research Article

The Theory for 𝐽-Hermitian Subspaces in a Product Space

Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, China

Received 6 January 2012; Accepted 13 February 2012

Academic Editors: S. Deng and O. Miyagaki

Copyright © 2012 Huaqing Sun and Jiangang Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, NY, USA, 1987.
  2. W. N. Everitt and L. Markus, Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators, vol. 61, American Mathematical Society, Providence, RI, USA, 1999.
  3. J. Weidmann, Spectral Theory of Ordinary Differential Operators, vol. 1258 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1987.
  4. S. Z. Fu, “On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces,” Journal of Differential Equations, vol. 100, no. 2, pp. 269–291, 1992. View at Publisher · View at Google Scholar
  5. J. Sun, “On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices,” Acta Mathematica Sinica, vol. 2, no. 2, pp. 152–167, 1986. View at Publisher · View at Google Scholar
  6. A. Wang, J. Sun, and A. Zettl, “Characterization of domains of self-adjoint ordinary differential operators,” Journal of Differential Equations, vol. 246, no. 4, pp. 1600–1622, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. Lesch and M. Malamud, “On the deficiency indices and self-adjointness of symmetric Hamiltonian systems,” Journal of Differential Equations, vol. 189, no. 2, pp. 556–615, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Y. Shi and H. Sun, “Self-adjoint extensions for second-order symmetric linear difference equations,” Linear Algebra and its Applications, vol. 434, no. 4, pp. 903–930, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. Arens, “Operational calculus of linear relations,” Pacific Journal of Mathematics, vol. 11, pp. 9–23, 1961. View at Zentralblatt MATH
  10. H. Langer and B. Textorius, “On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space,” Pacific Journal of Mathematics, vol. 72, no. 1, pp. 135–165, 1977.
  11. E. A. Coddington, Extension Theory of Formally Normal and Symmetric Subspaces, Memoirs of the American Mathematical Society, No. 134, American Mathematical Society, Providence, RI, USA, 1973.
  12. Y. Shi, “The Glazman-Krein-Naimark theory for Hermitian subspaces,” The Journal of Operator Theory. In press.
  13. E. A. Coddington, “Self-adjoint subspace extensions of nondensely defined symmetric operators,” Advances in Mathematics, vol. 14, pp. 309–332, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. E. A. Coddington and A. Dijksma, “Self-adjoint subspaces and eigenfunction expansions for ordinary differential subspaces,” Journal of Differential Equations, vol. 20, no. 2, pp. 473–526, 1976. View at Publisher · View at Google Scholar
  15. A. Dijksma and H. S. V. de Snoo, “Self-adjoint extensions of symmetric subspaces,” Pacific Journal of Mathematics, vol. 54, pp. 71–100, 1974. View at Zentralblatt MATH
  16. B. M. Brown, D. K. R. McCormack, W. D. Evans, and M. Plum, “On the spectrum of second-order differential operators with complex coefficients,” Proceedings of The Royal Society of London Series A, vol. 455, no. 1984, pp. 1235–1257, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. B. M. Brown and M. Marletta, “Spectral inclusion and spectral exactness for singular non-self-adjoint Sturm-Liouville problems,” Proceedings of The Royal Society of London Series A, vol. 457, no. 2005, pp. 117–139, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Qi, Z. Zheng, and H. Sun, “Classification of Sturm-Liouville differential equations with complex coefficients and operator realizations,” Proceedings of The Royal Society of London Series A, vol. 467, no. 2131, pp. 1835–1850, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A. R. Sims, “Secondary conditions for linear differential operators of the second order,” Journal of Mathematics and Mechanics, vol. 6, pp. 247–285, 1957. View at Zentralblatt MATH
  20. H. Sun and J. Qi, “On classification of second-order differential equations with complex coefficients,” Journal of Mathematical Analysis and Applications, vol. 372, no. 2, pp. 585–597, 2010. View at Publisher · View at Google Scholar
  21. H. Sun, J. Qi, and H. Jing, “Classification of non-self-adjoint singular Sturm-Liouville difference equations,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 8020–8030, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. R. H. Wilson, “Non-self-adjoint difference operators and their spectrum,” Proceedings of The Royal Society of London Series A, vol. 461, no. 2057, pp. 1505–1531, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. I. M. Glazman, “An analogue of the extension theory of Hermitian operators and a non-symmetric one-dimensional boundary problem on a half-axis,” Doklady Akademii Nauk SSSR, vol. 115, pp. 214–216, 1957 (Russian). View at Zentralblatt MATH
  24. D. Race, “The theory of J-self-adjoint extensions of J-symmetric operators,” Journal of Differential Equations, vol. 57, pp. 258–274, 1985.
  25. Z. J. Shang, “On J-self-adjoint extensions of J-symmetric ordinary differential operators,” Journal of Differential Equations, vol. 73, no. 1, pp. 153–177, 1988. View at Publisher · View at Google Scholar
  26. I. Knowles, “On the boundary conditions characterizing J-self-adjoint extensions of J-symmetric operators,” Journal of Differential Equations, vol. 40, no. 2, pp. 193–216, 1981. View at Publisher · View at Google Scholar
  27. J. L. Liu, “J self-adjoint extensions of J symmetric operators,” Acta Scientiarum Naturalium Universitatis Intramongolicae, vol. 23, no. 3, pp. 312–316, 1992.