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Maps Preserving the \(\partial \)-Spectrum of Product or Triple Product of Operators

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Abstract

Let X be an infinite-dimensional Banach space, and \(\mathcal {B}(X)\) be the algebra of all bounded linear operators on X. A map \(\Delta \), from \( \mathcal {B}(X) \) into a closed subsets of \( \mathbb {C}\ \) is said to be \( \partial \)-spectrum if \( \partial (\sigma (T))\subseteq \Delta (T) \subseteq \sigma (T) \) for all \(T \in \mathcal {B}(X)\). Here, \( \sigma (T) \) is spectrum of T and \( \partial (\sigma (T)) \) the boundary of \(\sigma (T)\). In this paper, we determine the forms of all surjective maps \( \phi \) from \( \mathcal {B}(X) \) into itself that satisfy either \( \Delta (\phi (T)\phi (S)) = \Delta ( TS) \) for all \(T, S \in \mathcal {B}(X)\) or \( \Delta (\phi (T)\phi (S)\phi (T)) = \Delta (TST) \) for all \( T, S\in \mathcal {B}(X) \).

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Acknowledgements

The authors wish to express their thanks to the referee for carefully reading the paper and for giving valuable suggestions.

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  1. Faculty of Sciences DharMahraz Fes, University Sidi Mohammed Ben Abdellah, Fez, Morocco

    Hassane Benbouziane, Aukacha Daoudi, Mustapha Ech-Chérif El Kettani & Ismail El Khchin

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  1. Hassane Benbouziane
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  2. Aukacha Daoudi
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  3. Mustapha Ech-Chérif El Kettani
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  4. Ismail El Khchin
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Correspondence to Hassane Benbouziane.

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Benbouziane, H., Daoudi, A., Kettani, M.EC.E. et al. Maps Preserving the \(\partial \)-Spectrum of Product or Triple Product of Operators. Mediterr. J. Math. 20, 312 (2023). https://doi.org/10.1007/s00009-023-02501-3

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  • DOI: https://doi.org/10.1007/s00009-023-02501-3

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