Solutions of Sylvester equation in C∗-modular operators

Authors

  • Z. Niazi Moghani Dep. Math., Mashhad Branch, Islamic Azad. Univ., Iran
  • M. Mohammadzadeh Karizaki Dep. Comput. Eng., Univ. Torbat Heydarieh, Iran
  • M. Khanehgir Dep. Math., Mashhad Branch, Islamic Azad. Univ., Iran

DOI:

https://doi.org/10.37863/umzh.v73i3.152

Keywords:

Hilbert C ∗ -module, Moore-Penrose inverse, Operator equation, Positive solution, Sylvester equation

Abstract

UDC 517.9

We study the solvability of the Sylvester equation AX+YB=C and the operator equation AXD+FYB=C in the general setting of the adjointable operators between Hilbert C∗ -modules. Based on the Moore – Penrose inverses of the associated operators, we propose necessary and sufficient conditions for the existence of solutions to these equations, and obtain the general expressions of the solutions in the solvable cases. We also provide an approach to the study of the positive solutions for a special case of Lyapunov equation.

References

A. A. Boichuk, A. A. Pokutnyi, Perturbation theory of operator equations in the Fr´echet and Hilbert spaces, Ukr. Math. J., 67, № 9, 1327 – 1335 (2016), https://doi.org/10.1007/s11253-016-1156-y DOI: https://doi.org/10.1007/s11253-016-1156-y

A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, 2nd Ed., De Gruyter, Berlin (2016), https://doi.org/10.1515/9783110378443 DOI: https://doi.org/10.1515/9783110378443

H. Braden, The equations ATX±XTA=B, SIAM J. Matrix Anal. and Appl., 20, 295 – 302 (1998), https://doi.org/10.1137/S0895479897323270 DOI: https://doi.org/10.1137/S0895479897323270

D. S. Cvetković-Ilić, J. J. Kohila, Positive and real-positive solutions to the equation axa∗=c in C∗ -algebras, Linear and Multilinear Algebra, 55, 535 – 543 (2007), https://doi.org/10.1080/03081080701248112 DOI: https://doi.org/10.1080/03081080701248112

A. Dajic, J. J. Koliha, Positive solutions equations AX=C and XB=D for Hilbert space operators, J. Math. Anal. and Appl., 333, 567 – 576 (2007), https://doi.org/10.1016/j.jmaa.2006.11.016 DOI: https://doi.org/10.1016/j.jmaa.2006.11.016

D. S. Djordjevic, Explicit solution of the operator equation A∗X+X∗A=B, J. Comput. and Appl. Math., 200, 701 – 704 (2007), https://doi.org/10.1016/j.cam.2006.01.023 DOI: https://doi.org/10.1016/j.cam.2006.01.023

G. R. Duan, The solution to the matrix equation AV+BW=EVJ+R, Appl. Math. Lett., 17, no. 10, 1197 – 1202 (2004), https://doi.org/10.1016/j.aml.2003.05.012 DOI: https://doi.org/10.1016/j.aml.2003.05.012

G. R. Duan, R. J. Patton, Robust fault detection in linear systems using Luenberger-type unknown input observers–a parametric approach, Int. G. Syst. Sci., 32, № 4, 533 – 540 (2001), https://doi.org/10.1080/002077201300080992 DOI: https://doi.org/10.1080/002077201300080992

X. Fang, J. Yu., Solutions to operator equations on Hilbert C∗ -modules, II, Integral Equat. and Oper. Theory 68, 23 – 60 (2010), https://doi.org/10.1007/s00020-010-1783-x DOI: https://doi.org/10.1007/s00020-010-1783-x

R. E. Harte, M. Mbekhta, On generalized inverses in C∗ -algebras, Stud. Math., 103, 71 – 77 (1992), https://doi.org/10.4064/sm-103-1-71-77 DOI: https://doi.org/10.4064/sm-103-1-71-77

C. G. Khatri, S. K. Mitra, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31, no. 4, 579 – 585 (1976), https://doi.org/10.1137/0131050 DOI: https://doi.org/10.1137/0131050

P. Kirrinnis, Fast algorithms for the Sylvester equation AXXBT=C, Theor. Comput. Sci., 259, 623 – 638 (2001), https://doi.org/10.1016/S0304-3975(00)00322-4 DOI: https://doi.org/10.1016/S0304-3975(00)00322-4

E. C. Lance, Hilbert C∗ -modules, LMS Lect. Note Ser. 210, Cambridge Univ. Press (1995), https://doi.org/10.1017/CBO9780511526206 DOI: https://doi.org/10.1017/CBO9780511526206

M. Mohammadzadeh Karizaki, M. Hassani, M. Amyari, M. Khosravi, Operator matrix of Moore – Penrose inverse operators on Hilbert C∗ -modules, Colloq. Math., 140, no. 2, 171 – 182 (2015), https://doi.org/10.4064/cm140-2-2 DOI: https://doi.org/10.4064/cm140-2-2

M. Mohammadzadeh Karizaki, M. Hassani, S. S. Dragomir, Explicit solution to modular operator equation TXS∗−SX∗T∗=A, Kragujevac J. Math., 40, № 2, 280 – 289 (2016), https://doi.org/10.5937/kgjmath1602280k DOI: https://doi.org/10.5937/KgJMath1602280K

T. Mor, N. Fukuma, M. Kuwahara, Explicit solution and eigenvalue bounds in the Lyapunov matrix equation, IEEE Trans. Automat. Control, 31, no. 7, 656 – 658 (1986), https://doi.org/10.1109/TAC.1986.1104369 DOI: https://doi.org/10.1109/TAC.1986.1104369

Z. Mousavi, R. Eskandari, M. S. Moslehian, F. Mirzapour, Operator equations AX+YB=C and AX+YB=C and AXA∗+BYB∗=C in Hilbert C∗-modules, Linear Algebra and Appl., 517, 85 – 98 (2017), https://doi.org/10.1016/j.laa.2016.12.001 DOI: https://doi.org/10.1016/j.laa.2016.12.001

F. Piao, Q. Zhanga, Z. Wang, The solution to matrix equation AX+XTC=B, J. Franklin Inst., 344, no. 8, 1056 – 1062 (2007), https://doi.org/10.1016/j.jfranklin.2007.05.002 DOI: https://doi.org/10.1016/j.jfranklin.2007.05.002

M. Wang, X. Cheng, M. Wei, Iterative algorithms for solving the matrix equation AXB+CXTD=E, Appl. Math. and Comput., 187, № 2, 622 – 629 (2007), https://doi.org/10.1016/j.amc.2006.08.169 DOI: https://doi.org/10.1016/j.amc.2006.08.169

Q. Xu, Common Hermitian and Positive solutions to the adjointable operator equations AX=C,XB=D, Linear Algebra and Appl., 429, no. 1, 1 – 11 (2008), https://doi.org/10.1016/j.laa.2008.01.030 DOI: https://doi.org/10.1016/j.laa.2008.01.030

Q. Xu, L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C∗ -modules, Linear Algebra and Appl., 428, 992 – 1000 (2008), https://doi.org/10.1016/j.laa.2007.08.035 DOI: https://doi.org/10.1016/j.laa.2007.08.035

Q. Xu, L. Sheng, Y. Gu, The solutions to some operator equations, Linear Algebra and Appl., 429, no. 8 - 9, 1997 – 2024 (2008), https://doi.org/10.1016/j.laa.2008.05.034 DOI: https://doi.org/10.1016/j.laa.2008.05.034

X. Zhang, Hermitian nonnegative-definite and positive-definite solutions of the matrix equation AXB=C, Appl. Math. E-Notes 4, 40 – 47 (2004).

B. Zhou, G. R. Duan, An explicit solution to the matrix equation AX−XF=BY, Linear Algebra and Appl., 402, 345 – 366 (2005), https://doi.org/10.1016/j.laa.2005.01.018 DOI: https://doi.org/10.1016/j.laa.2005.01.018

Downloads

Published

11.03.2021

Issue

Vol. 73 No. 3 (2021)

Section

Research articles

How to Cite

Moghani, Z. Niazi, et al. “Solutions of Sylvester Equation in C∗-Modular Operators”. Ukrains’kyi Matematychnyi Zhurnal, vol. 73, no. 3, Mar. 2021, pp. 354-69, https://doi.org/10.37863/umzh.v73i3.152.