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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
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[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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