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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.

47A62 Equations involving linear operators, with operator unknowns
46L08 \(C^*\)-modules
15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
Full Text: DOI
[1] Cvetković-Ilić, D.S.; Dajić, A.; Koliha, J.J., Positive and real-positive solutions to the equation \(\mathit{axa}^\ast = c\) 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 \(\mathit{AX} = C\) and \(\mathit{XB} = D\) for Hilbert space operators, J. math. anal. appl., 333, 567-576, (2007) · Zbl 1120.47009
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