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 $L_{\mathrm{\infty }}$-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 $T\mathrm{.}$ 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 ${\sum }_{k=1}^{\mathrm{\infty }}{\mu }_k(T)\mathrm{.}$ We show that for $p\in [1,\mathrm{\infty }]$ and $s\in (0,1]$with$1/s=1+|1/2-1/p|,$andforevery$s$-nuclearoperator$T$ in every subspace of any $L_p(\nu )$-space the trace of $T$ is well defined and equals the sum of all eigenvalues of $T\mathrm{.}$