In (Aiena et al., Math. Proc. R. Irish Acad. 122A(2):101–116, 2022), it has been shown that a bounded linear operator T ∈ L(X), defined on an infinite-dimensional complex Banach space X, for which there exists an injective quasi-nilpotent operator that commutes with it, has a very special structure of the spectrum. In this paper, we show that we have much more: if a such quasi-nilpotent operator does exist, then some of the spectra of T originating from B-Fredholm theory coalesce. Further, the spectral mapping theorem holds for all the B-Weyl spectra. Finally, the generalized version of Weyl type theorems hold for T assuming that T is of polaroid type. Our results apply to the operators that belong to the commutant of Volterra operators.

Aiena, P., Burderi, F., Triolo, S. (2024). Further Properties of an Operator Commuting with an Injective Quasi-Nilpotent Operator. MEDITERRANEAN JOURNAL OF MATHEMATICS, 21(1) [10.1007/s00009-023-02570-4].

Further Properties of an Operator Commuting with an Injective Quasi-Nilpotent Operator

Aiena, Pietro;Burderi, Fabio;Triolo, Salvatore
2024-01-13

Abstract

In (Aiena et al., Math. Proc. R. Irish Acad. 122A(2):101–116, 2022), it has been shown that a bounded linear operator T ∈ L(X), defined on an infinite-dimensional complex Banach space X, for which there exists an injective quasi-nilpotent operator that commutes with it, has a very special structure of the spectrum. In this paper, we show that we have much more: if a such quasi-nilpotent operator does exist, then some of the spectra of T originating from B-Fredholm theory coalesce. Further, the spectral mapping theorem holds for all the B-Weyl spectra. Finally, the generalized version of Weyl type theorems hold for T assuming that T is of polaroid type. Our results apply to the operators that belong to the commutant of Volterra operators.
13-gen-2024
Settore MAT/05 - Analisi Matematica
Aiena, P., Burderi, F., Triolo, S. (2024). Further Properties of an Operator Commuting with an Injective Quasi-Nilpotent Operator. MEDITERRANEAN JOURNAL OF MATHEMATICS, 21(1) [10.1007/s00009-023-02570-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/621520
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