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Aluthge Operator Field and Its Numerical Range and Spectral Properties

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Abstract

For an arbitrary operator T acting on a Hilbert space we consider a field of operators \(\left( \Delta _{z}(T)\right) \) called the Aluthge operator field associated with T. After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field \(\left( \Delta _{z}(T)\right) \) and a support subspace, are constant on each horizontal segment where they are defined. This result leads to a positive solution of a conjecture stated by Jung-Ko-Pearcy in 2000. Then we do a detailed spectral study of \(\left( \Delta _{z}(T)\right) \) and we give a Yamazaki type formula in this context.

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Authors and Affiliations

  1. Institut Camille Jordan CNRS UMR 5208, Université de Lyon 1, 43, boulevard du 11 Novembre 1918, 69622, Villeurbanne, France

    Gilles Cassier

  2. École normale supérieure Paris-Saclay, 4 Avenue des Sciences, 91190, Gif-sur-Yvette, France

    Thomas Perrin

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  1. Gilles Cassier
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  2. Thomas Perrin
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Correspondence to Gilles Cassier.

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Cassier, G., Perrin, T. Aluthge Operator Field and Its Numerical Range and Spectral Properties. Integr. Equ. Oper. Theory 93, 41 (2021). https://doi.org/10.1007/s00020-021-02656-2

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  • DOI: https://doi.org/10.1007/s00020-021-02656-2

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