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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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References:
 [1] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. appl. math., 17, 434-440, (1969) · Zbl 0265.15002 [2] Ben-Israel, Adi; Greville, T.N.R., Generalized inverses, theory and applications, (1974), Springer New York · Zbl 0305.15001 [3] Berberian, S.K., Baer $$\ast$$-rings, die grundlehren der mathematischen wissenschaften, band 195, (1972), Springer New York [4] Bhaskara Rao, K.P.S., The theory of generalized inverses over commutative rings, Algebra, logic and applications series, vol. 17, (2002), Taylor & Francis London · Zbl 0992.15003 [5] Campbell, S.L.; Meyer, C.D., Generalized inverses of linear transformations, (1979), Pitman London · Zbl 0417.15002 [6] Cecioni, F., Sopra alcune operazioni algebriche sulle matrici, Ann. scuola norm. sup. Pisa sci. fis. mat., 11, (1910) · JFM 41.0193.02 [7] Chu, K.-W.E., Symmetric solutions of linear matric equations by matrix decompositions, Linear algebra appl., 119, 35-50, (1989) · Zbl 0688.15003 [8] Cvetković-Ilić, D.S.; Djordjević, D.S.; Rakočević, V., Schur complements in $$C^\ast$$-algebras, Math. nachr., 278, 1-7, (2005) [9] Dai, Hua, On the symmetric solutions of linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009 [10] Dajić, Alegra; 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 [11] Don, F.J.H., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 93, 1-7, (1987) · Zbl 0622.15001 [12] Groß, J., Explicit solutions to the matrix inverse problem $$\mathit{AX} = B$$, Linear algebra appl., 289, 131-134, (1999) · Zbl 0941.15010 [13] Harte, R.E.; Mbekhta, M., On generalized inverses in $$C^\ast$$-algebras, Studia math., 103, 71-77, (1992) · Zbl 0810.46062 [14] Haverić, M., On solutions of a matrix equations system $$\mathit{AX} = B, \mathit{XD} = E$$, Matematicki vesnik, 36, 11-16, (1984) [15] Higham, N.J., The symmetric procrustes problem, Bit, 28, 133-143, (1988) · Zbl 0641.65034 [16] Hodges, J.H., The matrix equation $$\mathit{AX} = B$$ in a finite field, Amer. math. monthly, 63, 243-244, (1956) · Zbl 0070.01706 [17] Horn, R.A.; Sergeichuk, V.; Shaked-Monderer, N., Solution of linear matrix equations in a $${}^\ast$$congruence class, Electron. J. linear algebra, 174, 153-156, (2005) · Zbl 1092.15010 [18] 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 [19] Koliha, J.J.; Patrício, P., Elements of rings with equal spectral idempotents, J. aust. math. soc., 72, 137-150, (2002) · Zbl 0999.16025 [20] Koliha, J.J.; Rakočević, V., Range projections and the moore – penrose inverse in rings with involution, Linear and multilinear algebra, 55, 103-112, (2007) · Zbl 1116.46042 [21] Meng, Chunjun; Hu, Xiyan; Zhang, Lei, The skew-symmetric orthogonal solutions of the matrix equation $$\mathit{AX} = B$$, Linear algebra appl., 402, 303-318, (2005) · Zbl 1128.15301 [22] Mitra, S.K., The matrix equations $$\mathit{AX} = C, \mathit{XB} = D$$, Linear algebra appl., 59, 171-181, (1984) [23] Patrício, P.; Puystjens, R., About the von Neumann regularity of triangular block matrices, Linear algebra appl., 332-334, 485-502, (2001) · Zbl 1002.15001 [24] Peng, Zhenyun; Hu, Xiyan, The reflexive and anti-reflexive solutions of the matrix equation $$\mathit{AX} = B$$, Linear algebra appl., 375, 147-155, (2003) · Zbl 1050.15016 [25] Penrose, R., A generalized inverse for matrices, Proc. Cambridge philos. soc., 51, 406-413, (1955) · Zbl 0065.24603 [26] Phadke, S.V.; Thakare, N.K., Generalized inverses and operator equations, Linear algebra appl., 23, 191-199, (1979) · Zbl 0403.47005 [27] Porter, A.D., Solvability of the matric equation $$\mathit{AX} = B$$, Linear algebra appl., 13, 177-184, (1976) · Zbl 0333.15008 [28] Porter, A.D.; Mousouris, N., Ranked solutions of $$\mathit{AXC} = B$$ and $$\mathit{AX} = B$$, Linear algebra appl., 24, 217-224, (1979) · Zbl 0411.15009 [29] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York [30] Tian, Y.; Takane, Y., Schur complements and banachiewicz – schur forms, Electron. J. linear algebra, 13, 405-418, (2005) · Zbl 1095.15006 [31] Uhlig, F., On the matrix equation $$\mathit{AX} = B$$ with applications to the generators of a controllability matrix, Linear algebra appl., 85, 203-209, (1987) · Zbl 0612.15006 [32] Wu, L., The re-positive definite solution to the matrix inverse problem $$\mathit{AX} = B$$, Linear algebra appl., 174, 145-151, (1992) [33] Wu, L.; Cain, B., The re-nonnegative definite solutions to the matrix inverse problem $$\mathit{AX} = B$$, Linear algebra appl., 236, 137-146, (1996) · Zbl 0851.15004 [34] Xie, D.; Zhang, Lei; Hu, X., The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. comput. math., 18, 597-608, (2000) · Zbl 0966.15008 [35] Qingxiang Xu, Common Hermitian and positive solutions to the adjointable operator equations $$\mathit{AX} = C, \mathit{XB} = D$$, J. Math. Anal. Appl., in press. · Zbl 1153.47012
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