Open AccessSolutions of Sylvester equation in -modular operators
Author(s) -
Mahnaz Khanehgir,
Z. Niazi Moghani,
Mehdi Mohammadzadeh Karizaki
Publication year - 2021
Publication title -
ukraïnsʹkij matematičnij žurnal
Language(s) - English
Resource type - Journals
ISSN - 1027-3190
DOI - 10.37863/umzh.v73i3.152
Subject(s) - sylvester equation , mathematics , operator (biology) , operator theory , algebra over a field , lyapunov equation , modular design , spectral theorem , pure mathematics , lyapunov function , computer science , eigenvalues and eigenvectors , chemistry , repressor , quantum mechanics , nonlinear system , transcription factor , gene , operating system , biochemistry , physics
UDC 517.9We study the solvability of the Sylvester equation and the operator equation in the general setting of the adjointable operators between Hilbert -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.