A generalization of the Zariouh's property (gaz) through local spectral theory
Resumen
In this work, we present for the first time an in-depth study of the relationship between the upper semi-B-Fredholm spectrum and the left Drazin spectrum. This connection leads to the definition of a new spectral property, denoted as (ggaz), which generalizes the previously studied property (gaz). Through the framework of local spectral theory, we derive several characterizations of operators that satisfy the (ggaz) property. Moreover, we demonstrate that the set of operators fulfilling this property constitutes a Banach space, highlighting the structural significance of (ggaz) in operator theory.
Key words: Property (gaz), Property (ggaz), semi-B-Fredholm operator, Left-Drazin Operator, SVEP..
Citas
1. P. Aiena: Fredholm and local spectral theory II, with application to Weyl-type theorems. Springer Lecture Notes of Math no. 2235, (2018).
2. I. Fredholm: , Sur une classe d’´equations fonctionelles, Acta Math. (1903).
3. Aiena, P. Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004.
4. P. Aiena, F. Burderi, S. Triolo, Local spectral properties under conjugations, Mediterr. J. Math., 18 (2021), 1–20. DOI: https://doi.org/10.1007/s00009-021-01731-7
5. P. Aiena, E. Aponte, J. Guill´en. The Zariouh’s property (gaz) through localized SVEP. Mat. Vesnik, 2020, 72 (4), 314-326.
6. L.K. Saul, K.Q.Weinberger, J.H. Ham, F. Sha, and D.D. Lee: Spectral methods for dimensionality reduction. Semi-supervised learning. 3, (2006), 566-806.
7. Yuxin Chen, Yuejie Chi, Jianqing Fan and Cong Ma: Spectral methods for data science: A statistical perspective.Found. Trends Mach. Learn. 14 (5), (2021), 566-806.
8. E. Aponte, J. Sanabria, L. V´asquez, Perturbation theory for property (VE) and tensor product, Mathematics, 9 (2021), 2275. DOI: https://doi.org/10.3390/math9212775
9. E. Aponte, N. Jayanthi, D. Quiroz, P. Vasanthakumar, Tensor product of operators satisfying Zariouh’s property (gaz), and stability under perturbations, Axioms, 11 (2022), 225. DOI: https://doi.org/10.3390/axioms11050225
10. Aponte, E., Soto, J., and Rosas E. Study of the property (bz) using local spectral theory methods. Arab Journal of Basic and Applied Sciences, 30(1), 665–674, 2023.
11. Aponte, E. Property (az) through Topological Notions and Some Applications; Trans. A Razmadze Math. Institute, 2022, 176 (3).
12. M. Berkani: On a class of quasi-Fredholm operators. Integral Equations and Operator Theory 34 (1), (1999), 244-249.
13. M. Berkani: Index of B-Fredholm operators and generalization of a Weyl’s theorem, Proceding American Mathematical Society, vol. 130, 6, (2001), 1717-1723.
14. M. Berkani, M. Sarih: On semi-B-Fredholm operators, Glasgow Math. J. 43, (2001), 457- 465.
15. Ben Ouidren, K. and Zariouh, H. New approach to a-Weyl’s theorem and some preservation results. Rend. Circ. Mat. Palermo Series 2, 70, 819–833, 2021.
16. B. P. Duggal, Tensor products and property (w), Rend. Circ. Mat. Palerm., 60 (2011), 23–30. DOI: https://doi.org/10.1007/s12215-011-0023-9
17. S. Grabiner: Uniform ascent and descent of bounded operators Journal of Mathematical Society of Japan 34 (1982), 317-337.
18. H. Heuser: Functional Analysis. (1982), Marcel Dekker, New York.
19. C. S. Kubrusly, B. P. Duggal, On Weyl’s theorem for tensor products, Glasgow Math. J., 55 (2013), 139–144. DOI: https://doi.org/10.1017/S0017089512000407
20. C. S. Kubrusly, B. P. Duggal, On Weyl and Browder spectra of tensor products, Glasgow Math. J., 50 (2008), 289–302. DOI: https://doi.org/10.1017/S0017089508004205
21. J. Sanabria, L. V´asquez, C. Carpintero, E. Rosas, O. Garc´ıa. On strong variations of Weyl type theorems. Acta Math. Univ. Comen. (N.S.), 2017, 86(2), 345-356.
2. I. Fredholm: , Sur une classe d’´equations fonctionelles, Acta Math. (1903).
3. Aiena, P. Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004.
4. P. Aiena, F. Burderi, S. Triolo, Local spectral properties under conjugations, Mediterr. J. Math., 18 (2021), 1–20. DOI: https://doi.org/10.1007/s00009-021-01731-7
5. P. Aiena, E. Aponte, J. Guill´en. The Zariouh’s property (gaz) through localized SVEP. Mat. Vesnik, 2020, 72 (4), 314-326.
6. L.K. Saul, K.Q.Weinberger, J.H. Ham, F. Sha, and D.D. Lee: Spectral methods for dimensionality reduction. Semi-supervised learning. 3, (2006), 566-806.
7. Yuxin Chen, Yuejie Chi, Jianqing Fan and Cong Ma: Spectral methods for data science: A statistical perspective.Found. Trends Mach. Learn. 14 (5), (2021), 566-806.
8. E. Aponte, J. Sanabria, L. V´asquez, Perturbation theory for property (VE) and tensor product, Mathematics, 9 (2021), 2275. DOI: https://doi.org/10.3390/math9212775
9. E. Aponte, N. Jayanthi, D. Quiroz, P. Vasanthakumar, Tensor product of operators satisfying Zariouh’s property (gaz), and stability under perturbations, Axioms, 11 (2022), 225. DOI: https://doi.org/10.3390/axioms11050225
10. Aponte, E., Soto, J., and Rosas E. Study of the property (bz) using local spectral theory methods. Arab Journal of Basic and Applied Sciences, 30(1), 665–674, 2023.
11. Aponte, E. Property (az) through Topological Notions and Some Applications; Trans. A Razmadze Math. Institute, 2022, 176 (3).
12. M. Berkani: On a class of quasi-Fredholm operators. Integral Equations and Operator Theory 34 (1), (1999), 244-249.
13. M. Berkani: Index of B-Fredholm operators and generalization of a Weyl’s theorem, Proceding American Mathematical Society, vol. 130, 6, (2001), 1717-1723.
14. M. Berkani, M. Sarih: On semi-B-Fredholm operators, Glasgow Math. J. 43, (2001), 457- 465.
15. Ben Ouidren, K. and Zariouh, H. New approach to a-Weyl’s theorem and some preservation results. Rend. Circ. Mat. Palermo Series 2, 70, 819–833, 2021.
16. B. P. Duggal, Tensor products and property (w), Rend. Circ. Mat. Palerm., 60 (2011), 23–30. DOI: https://doi.org/10.1007/s12215-011-0023-9
17. S. Grabiner: Uniform ascent and descent of bounded operators Journal of Mathematical Society of Japan 34 (1982), 317-337.
18. H. Heuser: Functional Analysis. (1982), Marcel Dekker, New York.
19. C. S. Kubrusly, B. P. Duggal, On Weyl’s theorem for tensor products, Glasgow Math. J., 55 (2013), 139–144. DOI: https://doi.org/10.1017/S0017089512000407
20. C. S. Kubrusly, B. P. Duggal, On Weyl and Browder spectra of tensor products, Glasgow Math. J., 50 (2008), 289–302. DOI: https://doi.org/10.1017/S0017089508004205
21. J. Sanabria, L. V´asquez, C. Carpintero, E. Rosas, O. Garc´ıa. On strong variations of Weyl type theorems. Acta Math. Univ. Comen. (N.S.), 2017, 86(2), 345-356.