Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
inverse along an element; $\{e, 1, 3\}$-inverse; ${\{f, 1, 4}\}$-inverse; weighted Moore-Penrose inverse; $(v,w)$-$(b,c)$-inverse; $w$-core inverse; dual $v$-core inverse
inverse along an element; $\{e, 1, 3\}$-inverse; ${\{f, 1, 4}\}$-inverse; weighted Moore-Penrose inverse; $(v,w)$-$(b,c)$-inverse; $w$-core inverse; dual $v$-core inverse
Summary:
Let $R$ be a unital $\ast $-ring. For any $a,s,t,v,w\in R$ we define the weighted $w$-core inverse and the weighted dual $s$-core inverse, extending the $w$-core inverse and the dual $s$-core inverse, respectively. An element $a\in R$ has a weighted $w$-core inverse with the weight $v$ if there exists some $x\in R$ such that $awxvx=x$, $xvawa=a$ and $(awx)^*=awx$. Dually, an element $a\in R$ has a weighted dual $s$-core inverse with the weight $t$ if there exists some $y\in R$ such that $ytysa=y$, $asaty=a$ and $(ysa)^*=ysa$. Several characterizations of weighted $w$-core invertible and weighted dual $s$-core invertible elements are given when weights $v$ and $t$ are invertible Hermitian elements. Also, the relations among the weighted $w$-core inverse, the weighted dual $s$-core inverse, the $e$-core inverse, the dual $f$-core inverse, the weighted Moore-Penrose inverse and the $(v,w)$-$(b,c)$-inverse are considered.
References:
[1] Baksalary, O. M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58 (2010), 681-697. DOI 10.1080/03081080902778222 | MR 2722752 | Zbl 1202.15009
[2] Benítez, J., Boasso, E.: The inverse along an element in rings. Electron. J. Linear Algebra 31 (2016), 572-592. DOI 10.13001/1081-3810.3113 | MR 3578393 | Zbl 1351.15004
[3] Benítez, J., Boasso, E.: The inverse along an element in rings with an involution, Banach algebras and $C^*$-algebras. Linear Multilinear Algebra 65 (2017), 284-299. DOI 10.1080/03081087.2016.1183559 | MR 3577449 | Zbl 1361.15004
[4] Benítez, J., Boasso, E., Jin, H.: On one-sided $(b,c)$-inverses of arbitrary matrices. Electron. J. Linear Algebra 32 (2017), 391-422. DOI 10.13001/1081-3810.3487 | MR 3761550 | Zbl 1386.15016
[5] Cline, R. E.: An Application of Representation for the Generalized Inverse of a Matrix. MRC Technical Report 592. University of Wisconsin, Madison (1965).
[6] Drazin, M. P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65 (1958), 506-514. DOI 10.2307/2308576 | MR 0098762 | Zbl 0083.02901
[7] Drazin, M. P.: A class of outer generalized inverses. Linear Algebra Appl. 436 (2012), 1909-1923. DOI 10.1016/j.laa.2011.09.004 | MR 2889966 | Zbl 1254.15005
[8] Drazin, M. P.: Weighted $(b,c)$-inverses in categories and semigroups. Commun. Algebra 48 (2020), 1423-1438. DOI 10.1080/00927872.2019.1687712 | MR 4079318 | Zbl 1466.18003
[9] Ferreyra, D. E., Levis, F. E., Thome, N.: Revisiting the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41 (2018), 265-281. DOI 10.2989/16073606.2017.1377779 | MR 3777887 | Zbl 1390.15010
[10] Gao, Y., Chen, J.: Pseudo core inverses in rings with involution. Commun. Algebra 46 (2018), 38-50. DOI 10.1080/00927872.2016.1260729 | MR 3764841 | Zbl 1392.15005
[11] Hartwig, R. E., Luh, J.: A note on the group structure of unit regular ring elements. Pac. J. Math. 71 (1977), 449-461. DOI 10.2140/pjm.1977.71.449 | MR 0442018 | Zbl 0355.16005
[12] Jacobson, N.: The radical and semi-simplicity for arbitrary rings. Am. J. Math. 67 (1945), 300-320. DOI 10.2307/2371731 | MR 0012271 | Zbl 0060.07305
[13] Li, T., Chen, J.: Characterizations of core and dual core inverses in rings with involution. Linear Multilinear Algebra 66 (2018), 717-730. DOI 10.1080/03081087.2017.1320963 | MR 3779145 | Zbl 1392.15008
[14] Malika, S. B., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226 (2014), 575-580. DOI 10.1016/j.amc.2013.10.060 | MR 3144334 | Zbl 1354.15003
[15] Mary, X.: On generalized inverses and Green's relations. Linear Algebra Appl. 434 (2011), 1836-1844. DOI 10.1016/j.laa.2010.11.045 | MR 2775774 | Zbl 1219.15007
[16] Mary, X., Patrício, P.: The inverse along a lower triangular matrix. Appl. Math. Comput. 219 (2012), 886-891. DOI 10.1016/j.amc.2012.06.060 | MR 2981280 | Zbl 1287.15001
[17] Mary, X., Patrício, P.: Generalized inverses modulo $\mathcal H$ in semigroups and rings. Linear Multilinear Algebra 61 (2013), 1130-1135. DOI 10.1080/03081087.2012.731054 | MR 3175351 | Zbl 1383.15005
[18] Mosić, D., Deng, C., Ma, H.: On a weighted core inverse in a ring with involution. Commun. Algebra 46 (2018), 2332-2345. DOI 10.1080/00927872.2017.1378895 | MR 3778394 | Zbl 1427.16034
[19] Mosić, D., Stanimirović, P. S., Sahoo, J. K., Behera, R., Katsikis, V. N.: One-sided weighted outer inverses of tensors. J. Comput. Appl. Math. 388 (2021), Article ID 113293, 23 pages. DOI 10.1016/j.cam.2020.113293 | MR 4185119 | Zbl 1458.15010
[20] Penrose, R.: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51 (1955), 406-413. DOI 10.1017/S0305004100030401 | MR 0069793 | Zbl 0065.24603
[21] Prasad, K. M., Bapat, R. B.: The generalized Moore-Penrose inverse. Linear Algebra Appl. 165 (1992), 59-69. DOI 10.1016/0024-3795(92)90229-4 | MR 1149746 | Zbl 0743.15007
[22] Prasad, K. M., Mohana, K. S.: Core-EP inverse. Linear Multilinear Algebra 62 (2014), 792-802. DOI 10.1080/03081087.2013.791690 | MR 1306.15006 | Zbl 1306.15006
[23] Rakić, D. S., Dinčić, N. Č., Djordjević, D. S.: Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463 (2014), 115-133. DOI 10.1016/j.laa.2014.09.003 | MR 3262392 | Zbl 1297.15006
[24] Rao, C. R., Mitra, S. K.: Generalized Inverse of a Matrices and Its Application. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York (1971). MR 0338013 | Zbl 0236.15004
[25] Wang, H., Liu, X.: Characterizations of the core inverse and the core partial ordering. Linear Multilinear Algebra 63 (2015), 1829-1836. DOI 10.1080/03081087.2014.975702 | MR 3305012 | Zbl 1325.15002
[26] Zhu, H., Wang, Q.-W.: Weighted pseudo core inverses in rings. Linear Multilinear Algebra 68 (2020), 2434-2447. DOI 10.1080/03081087.2019.1585742 | MR 4171235 | Zbl 1459.16037
[27] Zhu, H., Wang, Q.-W.: Weighted Moore-Penrose inverses and weighted core inverses in rings with involution. Chin. Ann. Math., Ser. B 42 (2021), 613-624. DOI 10.1007/s11401-021-0282-5 | MR 4289196 | Zbl 1491.16040
[28] Zhu, H., Wu, L., Chen, J.: A new class of generalized inverses in semigroups and rings with involution. (to appear) in Comm. Algebra. DOI 10.1080/00927872.2022.2150771 | MR 4561472
[29] Zhu, H., Wu, L., Mosić, D.: One-sided $w$-core inverses in rings with an involution. (to appear) in Linear Multilinear Algebra. DOI 10.1080/03081087.2022.2035308 | MR 4577209
Search
Partner of
