The solutions to some operator equations in Hilbert $C^*$-module
الموضوعات :
M. Mohammadzadeh Karizaki
1
(Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)
M. Hassani
2
(Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)
الکلمات المفتاحية: Operator equation, Moore-Penrose inverse, Complemented submodule, Closed range, Hilbert C*-module,
ملخص المقالة :
In this paper, we state some results on product of operators with closed rangesand we solve the operator equation $TXS^*-SX^*T^*= A$ in the general setting of theadjointable operators between Hilbert $C^*$-modules, when $TS = 1$. Furthermore, by usingsome block operator matrix techniques, we find explicit solution of the operator equation$TXS^*-SX^*T^*= A$.
[1] T. Aghasizadeha and S. Hejazian, Maps preserving semi-Fredholm operators on Hilbert C*-modules, J. Math. Anal. Appl. 354 (2009), 625-629.
[2] H. Braden, The equations AT X ± XT A = B, SIAM J. Matrix Anal. Appl. 20 (1998), 295–302.
[3] D. S. Djordjevic, Explicit solution of the operator equation A∗X + X∗A = B, J. Comput. Appl. Math. 200 (2007) 701–704
[4] D. S. Djordjevic and N. C. Dincic, Reverse order law for the Moore-Penrose inverse, J. Math. Anal. Appl. 361 (2010) 252-261.
[5] M. Frank, Geometrical aspects of Hilbert C*-modules, Positivity 3 (1999), 215-243.
[6] M. Frank, Self-duality and C∗-reflexivity of Hilbert C∗-modules, Z. Anal. Anwendungen 9 (1990), 165-176.
[7] E. C. Lance, Hilbert C∗-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995.
[8] M. Mohammadzadeh Karizaki, M. Hassani, Explicit solution to the operator equation T XS∗ − SX∗T ∗ = A in Hilbert C∗-module,(Submited)
[9] M. Mohammadzadeh Karizaki, M. Hassani, M. Amyari and M. Khosravi, Operator matrix of Moore-Penrose inverse operators on Hilbert C∗-modules, to appear in Colloq. Math.
[10] K. Sharifi, B. Ahmadi Bonakdar, The reverse order law for Moore-Penrose inverses of operators on Hilbert C∗-modules, to appear in Bull. Iranian Math. Soc.
[11] Q. Xu and L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C*-modules, Linear Algebra Appl. 428 (2008), 992-1000.
[12] Q. Xu, L. Sheng, Y. Gu, The solutions to some operator equations, Linear Algebra Appl. 429 (2008) 1997- 2024.
[13] Y. Yuan, Solvability for a class of matrix equation and its applications, J, Nanjing Univ. (Math. Biquart.) 18 (2001) 221-227.
