Abstract
An extensive matrix class over a complex Banach algebra is considered from the point of view of its quasinilpontency, nilpotency, as well as Drazin and generalized Drazin invertibilities. The matrices belonging to the class are obtained by alterations of a \(2 \times 2\) anti-triangular matrix with one undetermined entry, and the general expression characterizing the class involves powers of the anti-triangular matrix. The main result of the paper determines those values of the powers for which the quasinilpontency of a matrix from the class is equivalent to the quasinilpontency of its undetermined entry. Similar result was established with respect to the nilpotency as well. The paper provides also related observations concerned with relationships between the Drazin and generalized Drazin invertibilities of the matrix and the entry. Though rooted in the work of Drazin published in the mid-twentieth century, the investigations carried out in the paper were inspired by and build upon recent results concerned with the Drazin invertibility.
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Personal communication. H. Zou, T. Li, Y. Wei, On the g\(\pi \)-Hirano invertibility in Banach algebras, (submitted).
Personal communication. H. Zou, T. Li, Y. Wei, On the g\(\pi \)-Hirano invertibility in Banach algebras, (submitted).
References
Koliha, J.J.: A generalized Drazin inverse. Glasgow Math. J. 38, 367–381 (1996)
Benjamin, R.A.M.: Continuity of Drazin and generalized Drazin inversion in Banach algebras, Master thesis, Faculty of Science, Stellenbosch University, South Africa (2013)
Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)
Mosić, D.: The generalized and pseudo \(n\)-strong Drazin inverses in rings. Linear Multilinear Algebra 69, 361–375 (2021)
Chen, H., Sheibani, M.: Generalized Hirano inverses in Banach algebras. Filomat 33, 6239–6249 (2019)
Castro-González, N., Dopazo, E.: Representations of the Drazin inverse for a class of block matrices. Linear Algebra Appl. 400, 253–269 (2005)
Castro-González, N., Dopazo, E., Robles, J.: Formulas for the Drazin inverse of special block matrices. Appl. Math. Comput. 174, 252–270 (2006)
Zou, H., Chen, J., Mosić, D.: The Drazin invertibility of an anti-triangular matrix over a ring. Stud. Sci. Math. Hung. 54, 489–508 (2017)
Cvetković-Ilić, D.S.: Some results on the (2, 2, 0) Drazin inverse problem. Linear Algebra Appl. 438, 4726–4741 (2013)
Chen, H., Sheibani, M.: New characterizations of g-Drazin inverse in a Banach algebra. arXiv:2009.02477v1
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This work was supported and founded by Kuwait University Research Grant No.[SM02/23].
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Cvetković-Ilić, D., Baksalary, O.M. & Zou, H. Inheritance of quasinilpontency and Drazin invertibility between a matrix and its entry. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 82 (2024). https://doi.org/10.1007/s13398-024-01583-2
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DOI: https://doi.org/10.1007/s13398-024-01583-2