The inverse along an element in rings

[EN] Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.Benítez López, J.; Boasso, E. (2016). The inverse along an element in rings. The Electronic Journal of Linear Algebra. 31:572-592. doi:10.13001/1081-3810.3113S5725923

The inverse along an element in rings with an involution, Banach algebras and C*-algebras

[EN] Properties of the inverse along an element in rings with an involution, Banach algebras and C*-algebras will be studied unifying known expressions concerning generalized inverses.This work was supported by the Universidad Politecnica de Valencia [SP20120474].Benítez López, J.; Boasso, E. (2017). The inverse along an element in rings with an involution, Banach algebras and C*-algebras. Linear and Multilinear Algebra. 65(2):284-299. doi:10.1080/03081087.2016.1183559S28429965

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

Diferenciabilidad en espacios de Banach

Esta Tesis se centra en el estudio de la diferenciabilidad de Funciones definidas sobre subconjuntos de espacios de Banach, en especial se estudian las funciones convexas y continuas y más concretamente la norma. Se demuestra la íntima relación entre los diferentes tipos de diferenciabilidad (Fréchet, Gâteaux, fuertemente subdiferenciable, bastante suave, ...) y la estructura topológica de los Espacioes de Banach donde están definidas las funciones (espacios de Asplund, separabilidad, el espacio dual no tiene subespacioes propios normantes, normas ásperas...) Se concluye la Tesis con el estudio de la relación entre las propiedades topológicas anteriormetne dichas y la inmersión de subconjuntos débil-* homeomorfos al conjunto ternario de Cantor en la esfera unidad del dual.Benítez López, J. (2000). Diferenciabilidad en espacios de Banach [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/5422Palanci

El sistema JPEG y la transformada coseno

En este artículo se introduce la base matemática del formato JPG usado en fotografía: la transformada coseno, la cual se basa en el álgebra matricial y en la transformada discreta de Fourier.Benítez López, J. (2018). El sistema JPEG y la transformada coseno. http://hdl.handle.net/10251/95411DE

El sistema de compresión JPEG. Un pequeño paseo por la transformada discreta de Fourier y la coseno

Este artículo revisa el sistema de compresión de imágenes JPEG. Este sistema se basa en la transformada discreta coseno, que a su vez, se basa en la transformada discreta de Fourier. Ambas transformadas se introducen de forma razonada y se explica el uso de la transformada discreta coseno al sistema de compresión de los archivos JPEGBenítez López, J. (2016). El sistema de compresión JPEG. Un pequeño paseo por la transformada discreta de Fourier y la coseno. Gaceta de la Real Sociedad Matemática Española. 19(1):25-45. http://hdl.handle.net/10251/84402S254519

Co-EP Banach algebra elements

[EN] In this work, given a unital Banach algebra A and a \in A such that a has a Moore-Penrose inverse a^+, it will be characterized when a^+ a - a a^+ is invertible. A particular subset of this class of objects wil also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will also be consider.The third author is supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.Benítez López, J.; Boasso, E.; Rakocevic, V. (2015). Co-EP Banach algebra elements. Banach Journal of Mathematical Analysis. 9(1):27-41. doi:10.15352/bjma/09-1-3S27419

On one-sided (B,C)-inverses of arbitrary matrices

[EN] In this article, one-sided $(b, c)$-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In addition, the $(b, c)$-inverse and the inverse along an element will be also researched in the context of rectangular matrices.Benítez López, J.; Boasso, E.; Jin, H. (2017). On one-sided (B,C)-inverses of arbitrary matrices. ELECTRONIC JOURNAL OF LINEAR ALGEBRA. 32:391-422. doi:10.13001/1081-3810.3487S3914223

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

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

[EN] Based on the conditions ab(2) = 0 and b pi(ab) is an element of A(d), we derive that (ab)(n), (ba)(n), and ab + ba are all generalized Drazin invertible in a Banach algebra A, where n is an element of N and a and b are elements of A. By using these results, some results on the symmetry representations for the generalized Drazin inverse of ab + ba are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b.This work was supported by the National Natural Science Foundation of China (grant number: 11361009, 61772006,11561015), the Special Fund for Science and Technological Bases and Talents of Guangxi (grant number: 2016AD05050, 2018AD19051), the Special Fund for Bagui Scholars of Guangxi (grant number: 2016A17), the High level innovation teams and distinguished scholars in Guangxi Universities (grant number: GUIJIAOREN201642HAO), the Natural Science Foundation of Guangxi (grant number: 2017GXNSFBA198053, 2018JJD110003), and the open fund of Guangxi Key laboratory of hybrid computation and IC design analysis (grant number: HCIC201607).Qin, Y.; Liu, X.; Benítez López, J. (2019). Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra. Symmetry (Basel). 11(1):1-9. https://doi.org/10.3390/sym11010105S19111González, N. C. (2005). Additive perturbation results for the Drazin inverse. Linear Algebra and its Applications, 397, 279-297. doi:10.1016/j.laa.2004.11.001Zhang, X., & Chen, G. (2006). The computation of Drazin inverse and its application in Markov chains. Applied Mathematics and Computation, 183(1), 292-300. doi:10.1016/j.amc.2006.05.076Castro-González, N., Dopazo, E., & Martínez-Serrano, M. F. (2009). On the Drazin inverse of the sum of two operators and its application to operator matrices. Journal of Mathematical Analysis and Applications, 350(1), 207-215. doi:10.1016/j.jmaa.2008.09.035Qiao, S., Wang, X.-Z., & Wei, Y. (2018). Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra and its Applications, 542, 101-117. doi:10.1016/j.laa.2017.03.014Stanimirovic, P. S., Zivkovic, I. S., & Wei, Y. (2015). Recurrent Neural Network for Computing the Drazin Inverse. IEEE Transactions on Neural Networks and Learning Systems, 26(11), 2830-2843. doi:10.1109/tnnls.2015.2397551Koliha, J. J. (1996). A generalized Drazin inverse. Glasgow Mathematical Journal, 38(3), 367-381. doi:10.1017/s0017089500031803Hartwig, R. E., Wang, G., & Wei, Y. (2001). Some additive results on Drazin inverse. Linear Algebra and its Applications, 322(1-3), 207-217. doi:10.1016/s0024-3795(00)00257-3Djordjević, D. S., & Wei, Y. (2002). Additive results for the generalized Drazin inverse. Journal of the Australian Mathematical Society, 73(1), 115-126. doi:10.1017/s1446788700008508Liu, X., Xu, L., & Yu, Y. (2010). The representations of the Drazin inverse of differences of two matrices. Applied Mathematics and Computation, 216(12), 3652-3661. doi:10.1016/j.amc.2010.05.016Yang, H., & Liu, X. (2011). The Drazin inverse of the sum of two matrices and its applications. Journal of Computational and Applied Mathematics, 235(5), 1412-1417. doi:10.1016/j.cam.2010.08.027Harte, R. (1992). On generalized inverses in C*-algebras. Studia Mathematica, 103(1), 71-77. doi:10.4064/sm-103-1-71-77Djordjevic, D. S., & Stanimirovic, P. S. (2001). On the Generalized Drazin Inverse and Generalized Resolvent. Czechoslovak Mathematical Journal, 51(3), 617-634. doi:10.1023/a:1013792207970Cvetković-Ilić, D. S., Djordjević, D. S., & Wei, Y. (2006). Additive results for the generalized Drazin inverse in a Banach algebra. Linear Algebra and its Applications, 418(1), 53-61. doi:10.1016/j.laa.2006.01.015Liu, X., Qin, X., & Benítez, J. (2016). New additive results for the generalized Drazin inverse in a Banach algebra. Filomat, 30(8), 2289-2294. doi:10.2298/fil1608289lMosić, D., Zou, H., & Chen, J. (2017). The generalized Drazin inverse of the sum in a Banach algebra. Annals of Functional Analysis, 8(1), 90-105. doi:10.1215/20088752-3764461González, N. C., & Koliha, J. J. (2004). New additive results for the g-Drazin inverse. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 134(6), 1085-1097. doi:10.1017/s0308210500003632Mosić, D. (2014). A note on Cline’s formula for the generalized Drazin inverse. Linear and Multilinear Algebra, 63(6), 1106-1110. doi:10.1080/03081087.2014.92296