The $w$-core–EP inverse in rings with involution

Authors

  • Dijana Mosić Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000 Niš, Serbia
  • Huihui Zhu School of Mathematics, Hefei University of Technology, Hefei 230009, People's Republic of China
  • Liyun Wu School of Mathematics, Hefei University of Technology, Hefei 230009, People's Republic of China

DOI:

https://doi.org/10.33044/revuma.3478

Abstract

The main goal of this paper is to present two new classes of generalized inverses in order to extend the concepts of the (dual) core–EP inverse and the (dual) $w$-core inverse. Precisely, we introduce the $w$-core–EP inverse and its dual for elements of a ring with involution. We characterize the (dual) $w$-core–EP invertible elements and develop several representations of the $w$-core–EP inverse and its dual in terms of different well-known generalized inverses. Using these results, we get new characterizations and expressions for the core–EP inverse and its dual. We apply the dual $w$-core–EP inverse to solve certain operator equations and give their general solution forms.

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References

O. M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 no. 5-6 (2010), 681–697.  DOI  MR  Zbl

J. Benítez and E. Boasso, The inverse along an element in rings with an involution, Banach algebras and $C^*$-algebras, Linear Multilinear Algebra 65 no. 2 (2017), 284–299.  DOI  MR  Zbl

J. Benítez, E. Boasso, and H. Jin, On one-sided $(b,c)$-inverses of arbitrary matrices, Electron. J. Linear Algebra 32 (2017), 391–422.  DOI  MR  Zbl

C. Coll, M. Lattanzi, and N. Thome, Weighted G-Drazin inverses and a new pre-order on rectangular matrices, Appl. Math. Comput. 317 (2018), 12–24.  DOI  MR  Zbl

G. Dolinar, B. Kuzma, J. Marovt, and B. Ungor, Properties of core-EP order in rings with involution, Front. Math. China 14 no. 4 (2019), 715–736.  DOI  MR  Zbl

M. P. Drazin, Pseudo-inverses in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506–514.  DOI  MR  Zbl

M. P. Drazin, A class of outer generalized inverses, Linear Algebra Appl. 436 no. 7 (2012), 1909–1923.  DOI  MR  Zbl

D. E. Ferreyra, F. E. Levis, and N. Thome, Maximal classes of matrices determining generalized inverses, Appl. Math. Comput. 333 (2018), 42–52.  DOI  MR  Zbl

D. E. Ferreyra, F. E. Levis, and N. Thome, Revisiting the core EP inverse and its extension to rectangular matrices, Quaest. Math. 41 no. 2 (2018), 265–281.  DOI  MR  Zbl

Y. Gao and J. Chen, Pseudo core inverses in rings with involution, Comm. Algebra 46 no. 1 (2018), 38–50.  DOI  MR  Zbl

Y. Gao, J. Chen, and Y. Ke, $*$-DMP elements in $*$-semigroups and $*$-rings, Filomat 32 no. 9 (2018), 3073–3085.  DOI  MR  Zbl

Y. Gao, J. Chen, and P. Patrício, Continuity of the core-EP inverse and its applications, Linear Multilinear Algebra 69 no. 3 (2021), 557–571.  DOI  MR  Zbl

Y. Ke, L. Wang, and J. Chen, The core inverse of a product and $2×2$ matrices, Bull. Malays. Math. Sci. Soc. 42 no. 1 (2019), 51–66.  DOI  MR  Zbl

I. I. Kyrchei, Determinantal representations of the core inverse and its generalizations with applications, J. Math. 2019 (2019), Art. ID 1631979, 13 pp.  DOI  MR  Zbl

I. I. Kyrchei, Determinantal representations of the weighted core-EP, DMP, MPD, and CMP inverses, J. Math. 2020 (2020), Art. ID 9816038, 12 pp.  DOI  MR  Zbl

L. Lebtahi and N. Thome, A note on $k$-generalized projections, Linear Algebra Appl. 420 no. 2-3 (2007), 572–575.  DOI  MR  Zbl

H. Ma and P. S. Stanimirović, Characterizations, approximation and perturbations of the core-EP inverse, Appl. Math. Comput. 359 (2019), 404–417.  DOI  MR  Zbl

H. Ma, P. S. Stanimirović, D. Mosić, and I. I. Kyrchei, Sign pattern, usability, representations and perturbation for the core-EP and weighted core-EP inverse, Appl. Math. Comput. 404 (2021), Paper No. 126247, 19 pp.  DOI  MR  Zbl

X. Mary, On generalized inverses and Green's relations, Linear Algebra Appl. 434 no. 8 (2011), 1836–1844.  DOI  MR  Zbl

X. Mary and P. Patrício, The inverse along a lower triangular matrix, Appl. Math. Comput. 219 no. 3 (2012), 886–891.  DOI  MR  Zbl

X. Mary and P. Patrício, Generalized inverses modulo $H$ in semigroups and rings, Linear Multilinear Algebra 61 no. 8 (2013), 1130–1135.  DOI  MR  Zbl

D. Mosić, Weighted core-EP inverse of an operator between Hilbert spaces, Linear Multilinear Algebra 67 no. 2 (2019), 278–298.  DOI  MR  Zbl

D. Mosić, Core-EP inverse in rings with involution, Publ. Math. Debrecen 96 no. 3-4 (2020), 427–443.  DOI  MR  Zbl

D. Mosić and D. S. Djordjević, The gDMP inverse of Hilbert space operators, J. Spectr. Theory 8 no. 2 (2018), 555–573.  DOI  MR  Zbl

P. Patrício and A. Veloso da Costa, On the Drazin index of regular elements, Cent. Eur. J. Math. 7 no. 2 (2009), 200–205.  DOI  MR  Zbl

R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413.  MR  Zbl

K. M. Prasad and K. S. Mohana, Core-EP inverse, Linear Multilinear Algebra 62 no. 6 (2014), 792–802.  DOI  MR  Zbl

K. M. Prasad and M. D. Raj, Bordering method to compute core-EP inverse, Spec. Matrices 6 (2018), 193–200.  DOI  MR  Zbl

K. M. Prasad, M. D. Raj, and M. Vinay, Iterative method to find core-EP inverse, Bull. Kerala Math. Assoc. 16 no. 1 (2018), 139–152.  MR

J. K. Sahoo, R. Behera, P. S. Stanimirović, V. N. Katsikis, and H. Ma, Core and core-EP inverses of tensors, Comput. Appl. Math. 39 no. 1 (2020), Paper No. 9, 28 pp.  DOI  MR  Zbl

H. Wang, Core-EP decomposition and its applications, Linear Algebra Appl. 508 (2016), 289–300.  DOI  MR  Zbl

M. Zhou, J. Chen, T. Li, and D. Wang, Three limit representations of the core-EP inverse, Filomat 32 no. 17 (2018), 5887–5894.  DOI  MR  Zbl

H. Zhu, J. Chen, P. Patrício, and X. Mary, Centralizer's applications to the inverse along an element, Appl. Math. Comput. 315 (2017), 27–33.  DOI  MR  Zbl

H. Zhu and P. Patrício, Characterizations for pseudo core inverses in a ring with involution, Linear Multilinear Algebra 67 no. 6 (2019), 1109–1120.  DOI  MR  Zbl

H. Zhu and L. Wu, A new class of partial orders, Algebra Colloq. 30 no. 4 (2023), 585–598.  DOI  MR  Zbl

H. Zhu, L. Wu, and J. Chen, A new class of generalized inverses in semigroups and rings with involution, Comm. Algebra 51 no. 5 (2023), 2098–2113.  DOI  MR  Zbl

H. Zhu, X. Zhang, and J. Chen, Generalized inverses of a factorization in a ring with involution, Linear Algebra Appl. 472 (2015), 142–150.  DOI  MR  Zbl

H. Zou, J. Chen, T. Li, and Y. Gao, Characterizations and representations of the inverse along an element, Bull. Malays. Math. Sci. Soc. 41 no. 4 (2018), 1835–1857.  DOI  MR  Zbl

H. Zou, J. Chen, and P. Patrício, Reverse order law for the core inverse in rings, Mediterr. J. Math. 15 no. 3 (2018), Paper No. 145, 17 pp.  DOI  MR  Zbl

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Published

2024-10-09

Issue

Vol. 67 No. 2 (2024)

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Article

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Copyright (c) 2024 Dijana Mosić, Huihui Zhu, Liyun Wu

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