Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE SOLUTIONS OF SOME OPERATOR EQUATIONS

  • Published : 2008.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we consider the solvability and describe the set of the solutions of the operator equations $AX+X^{*}C=B$ and $AXB+B^{*}X^{*}A^{*}=C$. This generalizes the results of D. S. Djordjevic [Explicit solution of the operator equation $A^{*}X+X^{*}$A=B, J. Comput. Appl. Math. 200(2007), 701-704].

Keywords

References

  1. A. Ben-Israel and T. N. E. Greville, Generalized Inverses, Theory and applications. Second edition. CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 15. Springer-Verlag, New York, 2003
  2. S. L. Campbell and C. D. Meyer Jr., Generalized Inverses of Linear Transformations, Corrected reprint of the 1979 original. Dover Publications, Inc., New York, 1991
  3. S. R. Caradus, Generalized Inverses and Operator Theory, Queen's Papers in Pure and Applied Mathematics, 50. Queen's University, Kingston, Ont., 1978
  4. D. S. Djordjevic, Explicit solution of the operator equation $A^{*}$X+$X^{*}$A = B, J. Comput. Appl. Math. 200 (2007), no. 2, 701-704 https://doi.org/10.1016/j.cam.2006.01.023
  5. R. E. Harte, Invertibility and singularity for bounded linear operators, Monographs and Textbooks in Pure and Applied Mathematics, 109. Marcel Dekker, Inc., New York, 1988
  6. P. Kirrinnis, Fast algorithms for the Sylvester equation AX - X$B^{T}$= C, Theoret. Comput. Sci. 259 (2001), no. 1-2, 623-638 https://doi.org/10.1016/S0304-3975(00)00322-4
  7. G. Kitagawa, An algorithm for solving the matrix equation X = FX$F^{T}$ + S , International Journal of Control, 25 (1977), no. 5, 745-753 https://doi.org/10.1080/00207177708922266
  8. Y. X. Peng, X. Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB = C, Appl. Math. Comput. 160 (2005), no. 3, 763-777 https://doi.org/10.1016/j.amc.2003.11.030
  9. Z. Y. Peng and X. Y. Hu, The reflexive and anti-reflexive solutions of the matrix equation AX = B, Linear Algebra Appl. 375 (2003), 147-155 https://doi.org/10.1016/S0024-3795(03)00607-4
  10. D. C. Sorensen and A. C. Antoulas, The Sylvester equation and approximate balanced reduction, Linear Algebra Appl. 351/352 (2002), 671-700

Cited by

  1. Maximization and minimization of the rank and inertia of the Hermitian matrix expression A-BX-(BX)* with applications vol.434, pp.10, 2011, https://doi.org/10.1016/j.laa.2010.12.010