It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear
operator in a Hilbert space is central-symmetric i 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 Zd-symmetry of spectra introduced
by him. We study α-nuclear operators generated by the tensor elements of socalled 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 Zd-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.