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Common spectral properties and \(\nu \)-convergence

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Abstract

In this paper we show that if \(\{T_n\}\) is a sequence of bounded linear operators on a complex Banach space X which \(\nu \)-converges to two different bounded linear operators T and U, then T and U have the same parts of the spectrum. In particular, we generalize the results of Sánchez-Perales and Djordjević (J Math Anal Appl 433:405–415, 2016) and of Ammar (Indag Math 28:424–435, 2017). We also investigate the spectral \(\nu \)-continuity for the surjective spectrum.

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Acknowledgements

The authors are indebted to the referee for several helpful remarks and suggestions on a former version of this paper. Namely his suggestions that led to a simplification of the proof of Theorem 3.2 and also his suggestions to add Theorem 3.5 and Corollary 3.6.

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Authors and Affiliations

  1. Department of Mathematics, Dhar El Mahraz Faculty of Science, Sidi Mohamed Ben Abdellah University, 30003, Fez, Morocco

    Soufiane Hadji & Hassane Zguitti

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  1. Soufiane Hadji
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  2. Hassane Zguitti
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Correspondence to Hassane Zguitti.

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Communicated by Miklós Palfia.

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Hadji, S., Zguitti, H. Common spectral properties and \(\nu \)-convergence. Banach J. Math. Anal. 18, 39 (2024). https://doi.org/10.1007/s43037-024-00350-0

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