Weighted w -core inverses in rings

Liyun Wu; Huihui Zhu

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 581-602
  • ISSN: 0011-4642

Abstract

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Let R be a unital * -ring. For any a , s , t , v , w ∈ 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 ∈ R has a weighted w -core inverse with the weight v if there exists some x ∈ R such that a w x v x = x , x v a w a = a and ( a w x ) * = a w x . Dually, an element a ∈ R has a weighted dual s -core inverse with the weight t if there exists some y ∈ R such that y t y s a = y , a s a t y = a and ( y s a ) * = y s a . 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.

How to cite

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Wu, Liyun, and Zhu, Huihui. "Weighted $w$-core inverses in rings." Czechoslovak Mathematical Journal 73.2 (2023): 581-602. <http://eudml.org/doc/299373>.

@article{Wu2023,
abstract = {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.},
author = {Wu, Liyun, Zhu, Huihui},
journal = {Czechoslovak Mathematical Journal},
keywords = {inverse along an element; $\lbrace e, 1, 3\rbrace $-inverse; $\{\lbrace f, 1, 4\}\rbrace $-inverse; weighted Moore-Penrose inverse; $(v,w)$-$(b,c)$-inverse; $w$-core inverse; dual $v$-core inverse},
language = {eng},
number = {2},
pages = {581-602},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weighted $w$-core inverses in rings},
url = {http://eudml.org/doc/299373},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Wu, Liyun
AU - Zhu, Huihui
TI - Weighted $w$-core inverses in rings
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 581
EP - 602
AB - 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.
LA - eng
KW - inverse along an element; $\lbrace e, 1, 3\rbrace $-inverse; ${\lbrace f, 1, 4}\rbrace $-inverse; weighted Moore-Penrose inverse; $(v,w)$-$(b,c)$-inverse; $w$-core inverse; dual $v$-core inverse
UR - http://eudml.org/doc/299373
ER -

References

top
  1. Baksalary, O. M., Trenkler, G., 10.1080/03081080902778222, Linear Multilinear Algebra 58 (2010), 681-697. (2010) Zbl1202.15009MR2722752DOI10.1080/03081080902778222
  2. Benítez, J., Boasso, E., 10.13001/1081-3810.3113, Electron. J. Linear Algebra 31 (2016), 572-592. (2016) Zbl1351.15004MR3578393DOI10.13001/1081-3810.3113
  3. Benítez, J., Boasso, E., 10.1080/03081087.2016.1183559, Linear Multilinear Algebra 65 (2017), 284-299. (2017) Zbl1361.15004MR3577449DOI10.1080/03081087.2016.1183559
  4. Benítez, J., Boasso, E., Jin, H., 10.13001/1081-3810.3487, Electron. J. Linear Algebra 32 (2017), 391-422. (2017) Zbl1386.15016MR3761550DOI10.13001/1081-3810.3487
  5. Cline, R. E., An Application of Representation for the Generalized Inverse of a Matrix, MRC Technical Report 592. University of Wisconsin, Madison (1965). (1965) 
  6. Drazin, M. P., 10.2307/2308576, Am. Math. Mon. 65 (1958), 506-514. (1958) Zbl0083.02901MR0098762DOI10.2307/2308576
  7. Drazin, M. P., 10.1016/j.laa.2011.09.004, Linear Algebra Appl. 436 (2012), 1909-1923. (2012) Zbl1254.15005MR2889966DOI10.1016/j.laa.2011.09.004
  8. Drazin, M. P., 10.1080/00927872.2019.1687712, Commun. Algebra 48 (2020), 1423-1438. (2020) Zbl1466.18003MR4079318DOI10.1080/00927872.2019.1687712
  9. Ferreyra, D. E., Levis, F. E., Thome, N., 10.2989/16073606.2017.1377779, Quaest. Math. 41 (2018), 265-281. (2018) Zbl1390.15010MR3777887DOI10.2989/16073606.2017.1377779
  10. Gao, Y., Chen, J., 10.1080/00927872.2016.1260729, Commun. Algebra 46 (2018), 38-50. (2018) Zbl1392.15005MR3764841DOI10.1080/00927872.2016.1260729
  11. Hartwig, R. E., Luh, J., 10.2140/pjm.1977.71.449, Pac. J. Math. 71 (1977), 449-461. (1977) Zbl0355.16005MR0442018DOI10.2140/pjm.1977.71.449
  12. Jacobson, N., 10.2307/2371731, Am. J. Math. 67 (1945), 300-320. (1945) Zbl0060.07305MR0012271DOI10.2307/2371731
  13. Li, T., Chen, J., 10.1080/03081087.2017.1320963, Linear Multilinear Algebra 66 (2018), 717-730. (2018) Zbl1392.15008MR3779145DOI10.1080/03081087.2017.1320963
  14. Malika, S. B., Thome, N., 10.1016/j.amc.2013.10.060, Appl. Math. Comput. 226 (2014), 575-580. (2014) Zbl1354.15003MR3144334DOI10.1016/j.amc.2013.10.060
  15. Mary, X., 10.1016/j.laa.2010.11.045, Linear Algebra Appl. 434 (2011), 1836-1844. (2011) Zbl1219.15007MR2775774DOI10.1016/j.laa.2010.11.045
  16. Mary, X., Patrício, P., 10.1016/j.amc.2012.06.060, Appl. Math. Comput. 219 (2012), 886-891. (2012) Zbl1287.15001MR2981280DOI10.1016/j.amc.2012.06.060
  17. Mary, X., Patrício, P., 10.1080/03081087.2012.731054, Linear Multilinear Algebra 61 (2013), 1130-1135. (2013) Zbl1383.15005MR3175351DOI10.1080/03081087.2012.731054
  18. Mosić, D., Deng, C., Ma, H., 10.1080/00927872.2017.1378895, Commun. Algebra 46 (2018), 2332-2345. (2018) Zbl1427.16034MR3778394DOI10.1080/00927872.2017.1378895
  19. Mosić, D., Stanimirović, P. S., Sahoo, J. K., Behera, R., Katsikis, V. N., 10.1016/j.cam.2020.113293, J. Comput. Appl. Math. 388 (2021), Article ID 113293, 23 pages. (2021) Zbl1458.15010MR4185119DOI10.1016/j.cam.2020.113293
  20. Penrose, R., 10.1017/S0305004100030401, Proc. Camb. Philos. Soc. 51 (1955), 406-413. (1955) Zbl0065.24603MR0069793DOI10.1017/S0305004100030401
  21. Prasad, K. M., Bapat, R. B., 10.1016/0024-3795(92)90229-4, Linear Algebra Appl. 165 (1992), 59-69. (1992) Zbl0743.15007MR1149746DOI10.1016/0024-3795(92)90229-4
  22. Prasad, K. M., Mohana, K. S., 10.1080/03081087.2013.791690, Linear Multilinear Algebra 62 (2014), 792-802. (2014) Zbl1306.15006MR1306.15006DOI10.1080/03081087.2013.791690
  23. Rakić, D. S., Dinčić, N. Č., Djordjević, D. S., 10.1016/j.laa.2014.09.003, Linear Algebra Appl. 463 (2014), 115-133. (2014) Zbl1297.15006MR3262392DOI10.1016/j.laa.2014.09.003
  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). (1971) Zbl0236.15004MR0338013
  25. Wang, H., Liu, X., 10.1080/03081087.2014.975702, Linear Multilinear Algebra 63 (2015), 1829-1836. (2015) Zbl1325.15002MR3305012DOI10.1080/03081087.2014.975702
  26. Zhu, H., Wang, Q.-W., 10.1080/03081087.2019.1585742, Linear Multilinear Algebra 68 (2020), 2434-2447. (2020) Zbl1459.16037MR4171235DOI10.1080/03081087.2019.1585742
  27. Zhu, H., Wang, Q.-W., 10.1007/s11401-021-0282-5, Chin. Ann. Math., Ser. B 42 (2021), 613-624. (2021) Zbl1491.16040MR4289196DOI10.1007/s11401-021-0282-5
  28. Zhu, H., Wu, L., Chen, J., 10.1080/00927872.2022.2150771, (to appear) in Comm. Algebra. MR4561472DOI10.1080/00927872.2022.2150771
  29. Zhu, H., Wu, L., Mosić, D., 10.1080/03081087.2022.2035308, (to appear) in Linear Multilinear Algebra. MR4577209DOI10.1080/03081087.2022.2035308

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