Existence Criteria and Expressions of the (b, c)-Inverse in Rings and Their Applications

[EN] Let R be a ring. Existence criteria for the (b, c)-inverse are given. We present explicit expressions for the (b, c)-inverse by using inner inverses. We answer the question when the (b, c)-inverse of a ¿ R is an inner inverse of a. As applications, we give a unified theory of some well-known results of the {1, 3}-inverse, the {1, 4}-inverse, the Moore¿Penrose inverse, the group inverse and the core inverse.The first author is grateful to the China Scholarship Council for giving him a scholarship for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Benítez López, J. (2018). Existence Criteria and Expressions of the (b, c)-Inverse in Rings and Their Applications. Mediterranean Journal of Mathematics. 15(1). https://doi.org/10.1007/s00009-017-1056-xS151Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Benítez, J., Boasso, E., Jin, H.W.: On one-sided $$(B,C)$$(B,C)-inverses of arbitrary matrices. Electron. J. Linear Algebra 32, 391–422 (2017). arXiv:1701.09054v1Boasso, E., Kantún-Montiel, G.: The $$(b,c)$$(b,c)-inverses in rings and in the Banach context. Mediterr. J. Math. 14, 112 (2017). https://doi.org/10.1007/s00009-017-0910-1BhaskaraRao, K.R.S.: The Theory of Generalized Inverses over Commutative Rings. Taylor and Francis, London (2002)Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Pitman, London (1979)Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)Drazin, M.P.: Left and right generalized inverses. Linear Algebra Appl. 510, 64–78 (2016)Green, J.A.: On the structure of semigroups. Ann. Math. 54(1), 163–172 (1951)Hartwig, R.E.: Block generalized inverses. Arch. Ration. Mech. Anal. 61, 197–251 (1976)Han, R.Z., Chen, J.L.: Generalized inverses of matrices over rings. Chin. Q. J. Math. 7(4), 40–49 (1992)Ke, Y.Y., Cvetković-Ilić, D.S., Chen, J.L., Višnjić J.: New results on $$(b, c)$$(b,c)-inverses. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2017.1301362Ke, Y.Y., Višnjić, J., Chen, J.L.: One-sided $$(b, c)$$(b,c)-inverses in rings (2016). arXiv:1607.06230v1Mary, X.: On generalized inverse and Green’s relations. Linear Algebra Appl. 434, 1836–1844 (2011)Mary, X., Patrício, P.: Generalized inverses modulo $$\cal{H}$$H in semigroups and rings. Linear Multilinear Algebra 61(8), 1130–1135 (2013)von Neumann, J.: On regular rings. Proc. Natl. Acad. Sci. USA 22(12), 707–713 (1936)Rakić, D.S.: A note on Rao and Mitra’s constrained inverse and Drazin’s (b, c) inverse. Linear Algebra Appl. 523, 102–108 (2017)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, 115–133 (2014)Rao, C.R., Mitra, S.K.: Generalized inverse of a matrix and its application. In: Proceedings of the Sixth Berkeley Symposium on Mathematics, Statistics and Probability, vol. 1, pp. 601–620. University of California Press, Berkeley (1972)Wei, Y.M.: A characterization and representation of the generalized inverse $$A^{(2)}_{T, S}$$AT,S(2) and its applications. Linear Algebra Appl. 280, 87–96 (1998)Wang, L., Chen, J.L., Castro-González, N.: Characterizations of the $$(b, c)$$(b,c)-inverse in a ring (2015). arXiv:1507.01446v1Xu, S.Z., Chen, J.L., Zhang, X.X.: New characterizations for core inverses in rings with involution. Front. Math. China 12(1), 231–246 (2017

EP Elements in Rings with Involution

[EN] Let R be a unital ring with involution. We first show that the EP elements in R can be characterized by three equations. Namely, let a. R, then a is EP if and only if there exists x. R such that (xa)* = xa, xa(2) = a and ax(2) = x. Any EP element in R is core invertible and Moore-Penrose invertible. We give more equivalent conditions for a core (Moore-Penrose) invertible element to be an EP element. Finally, any EP element is characterized in terms of the n-EP property, which is a generalization of the bi-EP property.This research is supported by the National Natural Science Foundation of China (No. 11771076). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; Benítez López, J. (2019). EP Elements in Rings with Involution. Bulletin of the Malaysian Mathematical Sciences Society. 42(6):3409-3426. https://doi.org/10.1007/s40840-019-00731-xS34093426426Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Benítez, J.: Moore–Penrose inverses and commuting elements of $C^{*}$-algebras. J. Math. Anal. Appl. 345(2), 766–770 (2008)Bhaskara Rao, K.P.S.: The Theory of Generalized Inverses Over Commutative Rings. Taylor and Francis, London (2002)Boasso, E.: On the Moore–Penrose inverse, EP Banach space operators, and EP Banach algebra elements. J. Math. Anal. Appl. 339(2), 1003–1014 (2008)Chen, W.X.: On EP elements, normal elements and partial isometries in rings with involution. Electron. J. Linear Algebra 23, 553–561 (2012)Drivaliaris, D., Karanasios, S., Pappas, D.: Factorizations of EP operators. Linear Algebra Appl. 429, 1555–1567 (2008)Hartwig, R.E.: Block generalized inverses. Arch. Retion. Mech. Anal. 61(3), 197–251 (1976)Hartwig, R.E., Spindelböck, K.: Matrices for which $A^*$ and $A^{\dagger }$ commute. Linear Multilinear Algebra 14(3), 241–256 (1983)Koliha, J.J., Patrício, P.: Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72(1), 137–152 (2002)Mosić, D., Djordjević, D.S., Koliha, J.J.: EP elements in rings. Linear Algebra Appl. 431, 527–535 (2009)Mosić, D., Djordjević, D.S.: New characterizations of EP, generalized normal and generalized Hermitian elements in rings. Appl. Math. Comput. 218, 6702–6710 (2012)Patrício, P., Puystjens, R.: Drazin–Moore–Penrose invertiblity in rings. Linear Algebra Appl. 389, 159–173 (2004)Rakić, D.S., Dinčić, Nebojša Č., Djordjević, D.S.: Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)von Neumann, J.: On regular rings. Proc. Natl. Acad. Sci. USA 22(12), 707–713 (1936

Projections for generalized inverses

[EN] Let R be a unital ring with involution. In Section 2, for given two core invertible elements a, b. R, we investigate mainly the absorption law for the core inverse in virtue of the equality of the projections aa and .In Section 3, we study several relations concerning the projections a a and bb , where a . a{1, 2, 4} and b . b{1, 2, 3}. Some well- known results are extended to the *- reducing ring case. As an application, EP elements in a *- reducing ring are considered.This research was supported by the National Natural Science Foundation of China [grant number 11371089]. The first author is grateful to China Scholarship Councilor giving him a scholarship for his further study in Universitat Politecnica de Valencia Spain.Xu, S.; Chen, C.; Benítez López, J. (2018). Projections for generalized inverses. Linear and Multilinear Algebra. 66(8):1593-1605. https://doi.org/10.1080/03081087.2017.1364339S1593160566

On the continuity and differentiability of the (dual) core inverse in C*-algebras

[EN] The continuity of the core inverse and the dual core inverse is studied in the setting of -algebras. Later, this study is specialized to the case of bounded Hilbert space operators and to complex matrices. In addition, the differentiability of these generalized inverses is studied in the context of -algebras.The third author is supported by the Scientific Research Foundation for doctorate programme at Huaiyin Institute of Technology [grant number Z301B18534]. The third author is grateful to China Scholarship Council for helping him pursue his further study at Universitat Politècnica de València, SpainBenítez López, J.; Boasso, E.; Xu, S. (2020). On the continuity and differentiability of the (dual) core inverse in C*-algebras. Linear and Multilinear Algebra. 68(4):686-709. https://doi.org/10.1080/03081087.2018.1516187S686709684Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Rakić, D. S., Dinčić, N. Č., & Djordjević, D. S. (2014). Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra and its Applications, 463, 115-133. doi:10.1016/j.laa.2014.09.003Rakić, D. S., Dinčić, N. Č., & Djordjević, D. S. (2014). Core inverse and core partial order of Hilbert space operators. Applied Mathematics and Computation, 244, 283-302. doi:10.1016/j.amc.2014.06.112Xu, S., Chen, J., & Zhang, X. (2016). New characterizations for core inverses in rings with involution. Frontiers of Mathematics in China, 12(1), 231-246. doi:10.1007/s11464-016-0591-2Drazin, M. P. (2012). A class of outer generalized inverses. Linear Algebra and its Applications, 436(7), 1909-1923. doi:10.1016/j.laa.2011.09.004Boasso, E., & Kantún-Montiel, G. (2017). The (b, c)-Inverse in Rings and in the Banach Context. Mediterranean Journal of Mathematics, 14(3). doi:10.1007/s00009-017-0910-1Harte, R. (1992). On generalized inverses in C*-algebras. Studia Mathematica, 103(1), 71-77. doi:10.4064/sm-103-1-71-77Harte, R. (1993). Generalized inverses in C*-algebras II. Studia Mathematica, 106(2), 129-138. doi:10.4064/sm-106-2-129-138Penrose, R. (1955). A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 51(3), 406-413. doi:10.1017/s0305004100030401Mbekhta, M. (1992). Conorme et Inverse Généralisé Dans Les C*-Algèbres. Canadian Mathematical Bulletin, 35(4), 515-522. doi:10.4153/cmb-1992-068-8Du, F., & Xue, Y. (2012). Perturbation analysis of A_{T,S}^(2) on Banach Spaces. The Electronic Journal of Linear Algebra, 23. doi:10.13001/1081-3810.1543Benítez, J., Cvetković-Ilić, D., & Liu, X. (2014). On the continuity of the group inverse in $C^*$-algebras. Banach Journal of Mathematical Analysis, 8(2), 204-213. doi:10.15352/bjma/1396640064Benítez, J., & Cvetković-Ilić, D. (2013). On the elements aa† and a†a in a ring. Applied Mathematics and Computation, 222, 478-489. doi:10.1016/j.amc.2013.07.015Koliha, J. J. (2001). Continuity and differentiability of the Moore-Penrose inverse in $C^*$-algebras. MATHEMATICA SCANDINAVICA, 88(1), 154. doi:10.7146/math.scand.a-14320Koliha, J. J., & Rakočević, V. (2004). On the Norm of Idempotents in $C^*$ -Algebras. Rocky Mountain Journal of Mathematics, 34(2). doi:10.1216/rmjm/1181069874Benítez, J., & Liu, X. (2012). On the continuity of the group inverse. Operators and Matrices, (4), 859-868. doi:10.7153/oam-06-55Douglas, R. G. (1966). On majorization, factorization, and range inclusion of operators on Hilbert space. Proceedings of the American Mathematical Society, 17(2), 413-413. doi:10.1090/s0002-9939-1966-0203464-1Boasso, E. (2009). Drazin spectra of Banach space operators and Banach algebra elements. Journal of Mathematical Analysis and Applications, 359(1), 48-55. doi:10.1016/j.jmaa.2009.05.036Rakić, D. S. (2017). A note on Rao and Mitra’s constrained inverse and Drazin’s (b,c) inverse. Linear Algebra and its Applications, 523, 102-108. doi:10.1016/j.laa.2017.02.025Stewart, G. W. (1969). On the Continuity of the Generalized Inverse. SIAM Journal on Applied Mathematics, 17(1), 33-45. doi:10.1137/011700

Generalized core inverses of matrices

[EN] n this paper, we introduce two new generalized inverses of matrices, namely, the -core inverse and the (j, m)-core inverse. The -core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler [1] and the core-EP inverse defined by Manjunatha Prasad and Mohana [10]. The (j, m)-core inverse of a complex matrix extends the notions of the core inverse and the DMP-inverse defined by Malik and Thome [9]. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.This research is supported by the National Natural Science Foundation of China (NO. 11771076 and No. 11471186). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; Benítez López, J.; Wang, D. (2019). Generalized core inverses of matrices. Miskolc Mathematical Notes (Online). 20(1):565-584. https://doi.org/10.18514/MMN.2019.2594S56558420