Xu, Qingxiang Common Hermitian and positive solutions to the adjointable operator equations \(AX = C\), \(XB = D\). (English) Zbl 1153.47012 Linear Algebra Appl. 429, No. 1, 1-11 (2008). 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. Reviewer: Khristo N. Boyadzhiev (Ada) Cited in 2 ReviewsCited in 46 Documents 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 Keywords:Hilbert \(C^{*}\)-module; Moore-Penrose inverse; inner inverse; Hermitian solution; positive solution Citations:Zbl 1120.47009 PDFBibTeX XMLCite \textit{Q. Xu}, Linear Algebra Appl. 429, No. 1, 1--11 (2008; Zbl 1153.47012) Full Text: DOI 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 [3] Eric Chu, K., Symmetric solutions of linear matrix equation by matrix decompositions, Linear Algebra Appl., 119, 35-50 (1989) · Zbl 0688.15003 [4] Henk Don, F., On the symmetric solutions of a linear matrix equation, Linear Algebra Appl., 93, 1-7 (1987) · Zbl 0622.15001 [5] Khatri, C. G.; Mitra, S. K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math., 31, 579-585 (1976) · Zbl 0359.65033 [6] Lance, E. C., Hilbert \(C^\ast \)-modules - A Toolkit for Operator Algebraists (1995), Cambridge University Press · Zbl 0822.46080 [7] Li, J., Positive semidefinite partitioned matrices and a linear matrix equation and its inverse problem, J. Math. Res. Exposition, 14, 25-34 (1994) · Zbl 0830.15008 [8] G.K. Pedersen, \( C^*\); G.K. Pedersen, \( C^*\) [9] Wang, G.; Wei, Y.; Qiao, S., Generalized Inverses: Theory and Computations (2004), Science Press: Science Press Beijing, New York [10] Xu, Q.; Sheng, L., Positive semi-definite matrices of adjointable operators on Hilbert \(C^\ast \)-modules, Linear Algebra Appl., 428, 992-1000 (2008) · Zbl 1142.47003 [11] Yuan, Y., On the symmetric solutions of matrix equation \((AX, XC) = (B, D)\) (Chinese), J. East China Shipbuilding Institute (Nat. Sci.), 15, 82-85 (2001) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.