The symmetry of a spectrum of nuclear operators in subspaces of $L_p$-spaces

It was proved in the paper [1] that the spectrum of a nuclear operator $A$ acting on a separable Hilbert space is central-symmetric if and only if $\operatorname{trace}A^{2n - 1} = 0$, $n \in \mathbb N$.
We prove:
Theorem. Let $Y$ be a subspace of a quotient (or a quotient of a subspace) of an $L_p$-space, $1\le p\le\infty$ and $T\in N_s(Y,Y)$ ($s$-nuclear), where $1/s=1+|1/2-1/p|$. The spectrum of $T$ is central-symmetric if and only if $\operatorname{trace}A^{2n - 1} = 0$, $n =1,2,\dots$ .
Remark. $T$ is $s$-nuclear, if $T$ admits a representation
$$ T=\sum_i \lambda_i y'_i\otimes y_i, $$
where $(\lambda_i)\in l_s,$ $(y'_i)$ and $(y_i)$ are bounded.