Grothendieck-Lidskii theorem for subspaces and factor spaces of L_p-spaces

In 1955, A. Grothendieck has shown that if the linear operator $T$ in a Banach subspace of an $L_\infty$-space is 2/3-nuclear then the trace of $T$ is well defined and is equal to the sum of all eigenvalues $\{\mu_k(T)\}$ of $T.$ V.B. Lidski\v{\i}, in 1959, proved his famous theorem on the coincidence of the trace of the $S_1$-operator in $L_2(\nu)$ with its spectral trace $\sum_{k=1}^\infty \mu_k(T).$ We show that for $p\in[1,\infty]$ and $s\in (0,1]$ with $1/s=1+|1/2-1/p|,$ and for every $s$-nuclear operator $T$ in every subspace of any $L_p(\nu)$-space the trace of $T$ is well defined and equals the sum of all eigenvalues of $T.$