TY - JOUR AU - Z. Niazi Moghani AU - M. Mohammadzadeh Karizaki AU - M. Khanehgir PY - 2021/03/11 Y2 - 2024/03/29 TI - Solutions of Sylvester equation in $C^*$-modular operators JF - Ukrains’kyi Matematychnyi Zhurnal JA - Ukr. Mat. Zhurn. VL - 73 IS - 3 SE - Research articles DO - 10.37863/umzh.v73i3.152 UR - https://umj.imath.kiev.ua/index.php/umj/article/view/152 AB - UDC 517.9We 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. ER -