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Common Hermitian and positive solutions to the adjointable operator equations \(AX = C\), \(XB = D\). (English) Zbl 1153.47012

The author extends and corrects some results from [A.Dajić and J.J.Koliha, J. Math, Anal.Appl.333, No.2, 567–576 (2007; Zbl 1120.47009)], where positive common solutions \(X\) were found for the equations in the title within the framework of \(C^*\)-algebras. Working in the more general setting of Hilbert \(C^*\)-modules, the author of the present paper provides necessary and sufficient conditions for the existence of common Hermitian and positive solutions \(X\) to the above equations.

MSC:

47A62 Equations involving linear operators, with operator unknowns
46L08 \(C^*\)-modules
15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities

Citations:

Zbl 1120.47009
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References:

[1] Cvetković-Ilić, D. S.; Dajić, A.; Koliha, J. J., Positive and real-positive solutions to the equation \(axa^\ast = c\) in \(C^\ast \)-algebras, Linear and Multilinear Algebra, 55, 535-543 (2007) · Zbl 1180.47014
[2] Dajić, A.; Koliha, J. J., Positive solutions to the equations \(AX = C\) and \(XB = D\) for Hilbert space operators, J. Math. Anal. Appl., 333, 567-576 (2007) · Zbl 1120.47009
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