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Some new results on property \((UW {\scriptstyle \Pi })\)

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Abstract

Property \((UW {\scriptstyle \Pi })\) for a bounded linear operator \(T\in L(X)\) on a Banach space X means that the points \(\lambda \) of the approximate point spectrum for which \(\lambda I-T\) is upper semi-Weyl are exactly the poles of the resolvent of T. In this paper we give some new results on this property and relate it with some other variants of Browder’s theorem, as property (gaz) and property \((Z{\scriptstyle \Pi _a})\). In the last part the theory is applied to Toeplitz operators \(T_\phi \) with symbol \(\phi \) on Hardy spaces.

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Acknowledgements

The authors thank to Robin Harte for their careful reading of the paper and for drawing to their attention the fact that Theorem 2.5 of [5] was not true. We also thank the referee for several suggestions which improved the final version of this paper.

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Authors and Affiliations

  1. Dipartimento di Metodi e Modelli matematici, Facoltá di Ingegneria Universitá di Palermo, Palermo, Italy

    P. Aiena

  2. Department of Mathematics, Faculty of Sciences and Technology, Moulay Ismail University, Errachidia, Morocco

    M. Kachad

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  1. P. Aiena
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  2. M. Kachad
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Correspondence to P. Aiena.

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Aiena, P., Kachad, M. Some new results on property \((UW {\scriptstyle \Pi })\). Rend. Circ. Mat. Palermo, II. Ser 72, 591–604 (2023). https://doi.org/10.1007/s12215-021-00641-7

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  • DOI: https://doi.org/10.1007/s12215-021-00641-7

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