Equations \(ax = c\) and \(xb = d\) in rings and rings with involution with applications to Hilbert space operators (Q947645)

The authors review the equations \(ax = c\) and \(xb = d\) in the setting of associative rings with or without involution. The study of common solutions of the equations above in the framework of matrices dates back to the early 20th century; see \textit{F. Cecioni} [``Sopra alcune operazioni algebriche sulle matrici'' (Pisa Ann.\ 11) (1910; JFM 41.0193.02)]. The authors give necessary and sufficient conditions for the existence of the Hermitian, skew-Hermitian, reflexive, antireflexive, positive and real-positive solutions, and describe the general solutions in terms of the original elements or operators.