[Authors]
 O. I. Reinov

[Title]
 On $Z_d$-symmetry of spectra of some nuclear operators

[AMS Subj-class]
 47B06 Riesz operators; eigenvalue distributions; approximation numbers,
       s-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
 47B10 Operators belonging to operator ideals (nuclear, $p$-summing,
       in the Schatten-von Neumann classes, etc.)

[Abstract]
 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 $\Bbb Z_d$-symmetry of spectra
 introduced by him. We study $\al$-nuclear operators generated by the tensor
 elements of so-called $\al$-projective tensor products of Banach spaces,
 introduced in the paper ($\al$ is a quasi-norm). We give exact generalizations
 of Zelikin's theorem to the cases of $\Bbb Z_d$-symmetry of spectra of
 $\al$-nuclear operators (in particular, for $s$-nuclear and for $(r,p)$-nuclear
 operators). We show that the results are optimal.

[Comments]
 LaTeX, English, 15 pp.

[Contact e-mail]
 orein51@mail.ru