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Published in:
2020 | OriginalPaper | Chapter
On -symmetry of spectra of some nuclear operators
Author : Oleg Reinov
Published in: Analysis as a Tool in Mathematical Physics
Publisher: Springer International Publishing
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It was shown by M.I. Zelikin (2007) that the spectrum of a nuclear operator in a Hilbert space is central-symmetric iff the traces of all odd powers of the operator equal zero. B. Mityagin (2016) generalized Zelikin’s criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear, considering even a notion of so-called $$ \mathbb Z_d $$ -symmetry of spectra introduced by him. We study α-nuclear operators generated by the tensor elements of so-called α-projective tensor products of Banach spaces, introduced in the paper (α is a quasi-norm). We give exact generalizations of Zelikin’s theorem to the cases of $$ \mathbb Z_d $$ -symmetry of spectra of α-nuclear operators (in particular, for s-nuclear and for (r, p)-nuclear operators). We show that the results are optimal.