We introduce a new property (gaR) extending the property (R) considered by Aiena. We study the property (gaR) in connection with the Weyl-type theorems and establish sufficient and necessary conditions under which the property (gaR) holds. In addition, we also study the stability of the property (gaR) under perturbations by finite-dimensional operators, by nilpotent operators, by quasinilpotent operators, and by algebraic operators commuting with T. The classes of operators are considered as illustrating examples.
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References
P. Aiena, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer AP, New York (2004).
P. Aiena, E. Aponte, J. R. Guillén, and P. Peña, “Property (R) under perturbations,” Mediterran. J. Math., 10, No. 1, 367–382 (2013).
P. Aiena, C. Carpintero, and E. Rosas, “Some characterizations of operators satisfying a-Browder’s theorem,” J. Math. Anal. Appl., 311, No. 2, 530–544 (2005).
P. Aiena and O. Garcia, “Generalized Browder’s theorem and SVEP,” Mediterran. J. Math., 4, No. 2, 215–228 (2007).
P. Aiena, J. R. Guillén, and P. Peña, “Property (R) for bounded linear operators,” Mediterran. J. Math., 8, No. 4, 491–508 (2011).
P. Aiena and P. Peña, “Variations on Weyl’s theorem,” J. Math. Anal. Appl., 324, 566–579 (2006).
M. Amouch and M. Berkani, “On the property (gw),” Mediterran. J. Math., 5, No. 3, 371–378 (2008).
M. Amouch and H. Zguitti, “On the equivalence of Browder’s and generalized Browder’s theorem,” Glasgow Math. J., 48, No. 1, 179–185 (2006).
M. Berkani and J. J. Koliha, “Weyl type theorems for bounded linear operators,” Acta Sci. Math. (Szeged), 69, No. 1-2, 359–376 (2003).
M. Berkani and M. Sarih, “On semi B-Fredholm operators,” Glasgow Math. J., 43, No. 3, 457–465 (2001).
M. Berkani and H. Zariouh, “Extended Weyl type theorems,” Math. Bohem., 134, No. 4, 369–378 (2009).
M. Berkani and H. Zariouh, “New extended Weyl type theorems,” Mat. Vestnik, 62, No. 2, 145–154 (2010).
D. S. Djordjević, “Operators obeying a-Weyl’s theorem,” Publ. Math. Debrecen, 55, No. 3-4, 283–298 (1999).
S. Grabiner, “Uniform ascent and descent of bounded operators,” J. Math. Soc. Japan, 34, No. 2, 317–337 (1982).
H. Heuser, Functional Analysis, Wiley , New York (1982).
K. B. Laursen and M. M. Neumann, Introduction to Local Spectral Theory, Clarendon Press, Oxford (2000).
M. H. M. Rashid, “Properties (S) and (gS) for bounded linear operators,” Filomat, 28, No. 8, 1641–1652 (2014).
J. L. Shen and A. Chen, “A new variation of Weyl type theorem and perturbations,” Filomat, 28, No. 9, 1899–1906 (2014).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 7, pp. 974–983, July, 2017.
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Shen, J.L., Chen, A. A Note on the Property (gaR) and Perturbations. Ukr Math J 69, 1132–1143 (2017). https://doi.org/10.1007/s11253-017-1420-9
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DOI: https://doi.org/10.1007/s11253-017-1420-9