Solutions of Sylvester equation in $C^*$-modular operators

  • 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
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 $AX + Y B = C$ and the operator equation $AXD + FY B = C$ 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.

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Published
11.03.2021
How to Cite
Moghani, Z. N., M. Mohammadzadeh Karizaki, and M. Khanehgir. “Solutions of Sylvester Equation in $C^*$-Modular Operators”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 3, Mar. 2021, pp. 354 -69, doi:10.37863/umzh.v73i3.152.
Section
Research articles