  It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator
in a separable Hilbert space is central-symmetric iff the spectral traces of
all odd powers of the operator equal zero. The criterium can not be extended to
the case of general Banach spaces: It follows from Grothendieck-Enflo results
that there exists a nuclear operator $U$ in the space $l_1$ with the property
that $\operatorname{trace}\, U=1$ and $U^2=0.$ B. Mityagin (2016) has
generalized Zelikin's criterium to the case of compact operators (in Banach
spaces) some of which powers are nuclear. We give sharp generalizations of
Zelikin's theorem (to the cases of subspaces of quotients of $L_p$-spaces) and
of Mityagin's result (for the case where the operators are not necessarily
compact).
