Abstract
In this paper the notions of left and right generalized Drazin–Riesz invertible linear relations are introduced and studied. For these classes of linear relations we give several characterizations which covered the ones proved in Zivkovic-Zlatanovic and Cvetkovic (Linear Multilinear Algebra 65(6):1171–1193, 2017) in the particular case of bounded operators. Some characterizations are essentially expressed by means of a Kato–Riesz decomposition and an associated projection. Among other things, we show that these linear relations are completely characterized in terms of an algebraic decomposition with a semi-Browder linear relation and a bounded Riesz operator. Finally, the application of some obtained results is given.
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References
Abad, O., Zguitti, H.: A note on the generalized Drazin-Riesz invertible operators. Ann. Funct. Anal. 12(4), Paper No. 55, 17pp, (2021)
Aiena, P.: Fredholm and Local Spectral Theory with Applications to Multipliers. Kluwer Academic Publishers (2004)
Álvarez, T., Keskes, S., Mnif, M.: On the structrure of essentially semi-regular linear relations, Mediterr. J. Math. 16,76, 20pp (2019)
Álvarez, T., Sandovici, A.: Regular linear relations on Banach spaces. Banach J. Math. Anal. 15(4), 26 (2021)
Ammar, A., Bouchekoua, A., Jeribi, A.: The local spectral theory for linear relations involving SVEP. Mediterr. J. Math. 18(77), 27 (2021)
Benharrat, M., Miloud Hocine, K., Messirdi, B.: Left and right generalized Drazin invertible operators. Linear Multilinear Algebra 63(8), 1635–1648 (2015)
Benharrat, M., Miloud Hocine, K., Messirdi, B.: Left and right generalized Drazin invertible operators and local spectral theory. Proyecciones 38, 897–919 (2019)
Chamkha, Y., Mnif, M.: Browder spectra of upper triangular matrix linear relations. Publ. Math. Debr. 82(3–4), 569–590 (2013)
Cross, R.W.: Multivalued Linear Operator. Pure and Applied Mathematics, Marcel Dekker Inc (1998)
Cvetkovic, M.D.: On upper and lower generalized Drazin invertible operators. Funct. Anal. Approx. Comput. 7, 67–74 (2015)
Cvetkovic, M.D., Zivkovic-Zlatonovic, S.C.: Generalized Kato decomposition and essential spectra. Complex Anal. Oper. Theory 11(6), 1425–1449 (2017)
Drazin, M.P.: Left and right generalized inverses. Linear Algebra Appl. 510, 64–78 (2016)
Fakhfakh, F.: Perturbation results for some classes related to Browder linear relations and applications. Complex Anal. Oper. Theory 12, 2003–2018 (2018)
Fakhfakh, F., Mnif, M.: Perturbation theory for lower semi-Browder multivalued linear operators. Publ. Math. Debr. 78, 595–606 (2011)
Karmouni, M., Tajmouati, A.: A new characterization of Browder’s theorem. Filomat 32(14), 4865–4873 (2018)
Keskes, S.: Weyl’s type theorems for linear relations satisfying the single valued extension property. Monatsh. Math. 201, 803–824 (2023)
Lajnef, M., Mnif, M.: Upper and lower generalized Drazin invertible linear relations. Anal. Math. 48, 779–801 (2022)
Messirdi, So., Messirdi, Sa., Messirdi, B.: Further results on left and right generalized Drazin invertible operators. Mat. Stud. 54(1), 98–106 (2020)
Muller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Birkhauser, Switzerland (2007)
Ounadjela, D., Miloud Hocine, K., Messirdi, B.: The Perturbation Classes Problem for generalized Drazin invertible operators I. Rend. Circ. Math. Palermo (2) 67, 159–172 (2018)
Ounadjela, D., Benharrat, M., Messirdi, B.: The Perturbation Classes Problem for generalized Drazin invertible operators II. Complex Anal. Oper. Theory 13, 1361–1375 (2019)
Sandovici, A., de Snoo, H.S.V., Winker, H.: Ascent, descent, nullity, defect and related notions of linear relations in linear spaces. Linear Algebra Appl. 423, 456–497 (2007)
Sandovici, A., de Snoo, H.S.V.: An index formula for the product of linear relations. Linear Algebra Appl. 431, 2160–2171 (2009)
Zivkovic-Zlatanovic, S.C., Cvetkovic, M.D.: Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear Algebra 65(6), 1171–1193 (2017)
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Lajnef, M. Left and right generalized Drazin–Riesz invertible multivalued linear operators. Rend. Circ. Mat. Palermo, II. Ser 73, 1637–1652 (2024). https://doi.org/10.1007/s12215-024-01002-w
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DOI: https://doi.org/10.1007/s12215-024-01002-w
Keywords
- Left and right generalized Drazin–Riesz invertible linear relations
- SVEP
- Riesz operator
- Kato–Riesz decomposition