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Equations \(ax = c\) and \(xb = d\) in rings and rings with involution with applications to Hilbert space operators. (English) Zbl 1149.47011
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 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.

MSC:
47A62 Equations involving linear operators, with operator unknowns
15A24 Matrix equations and identities
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16B99 General and miscellaneous
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