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The local reduced minimum modulus on a Hilbert space

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Abstract

Let H be a complex Hilbert space and let \({\mathcal {B}}(H)\) be the algebra of all bounded linear operators on H. In this paper, for \(T\in {\mathcal {B}}(H)\) and a unit vector \(x\in H\), we introduce a local version of the reduced minimum modulus of T at x, noted by \(\gamma (T, x)\). Properties of this quantity are investigated. We study the relations between \(\gamma (T, x)\) and the Moore–Penrose inverse, spectrum of \(\vert T\vert \) and the local spectrum of \(\vert T\vert \) at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory).

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Acknowledgements

I would like to express my gratitude to anonymous referee for carefully reading the paper. His comments and suggestions greatly improved the final version of this article.

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Authors and Affiliations

  1. Mohammed V University in Rabat, Mathematics Departement, Rabat, Morocco

    Mostafa Mbekhta

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  1. Mostafa Mbekhta
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Correspondence to Mostafa Mbekhta.

Additional information

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and by Fincome program 470/2022 from CNRST- Morocco.

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Mbekhta, M. The local reduced minimum modulus on a Hilbert space. Acta Sci. Math. (Szeged) 89, 269–292 (2023). https://doi.org/10.1007/s44146-023-00060-3

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