The absence of local unconditional structure in some spaces of operators

We prove the following.
Theorem 1. {\it Let Banach spaces $X_n$, $Y_m$ have monotone unconditional bases $\{x_i\}_{i=1}^n$, $\{y_j\}_{j=1}^m$ resp., let $[\mathfrak A,\alpha]$ be a Banach ideal of operators, $S\in L(X_n,Y_m)$, $y'_j(Sx_i)=\pm1$. Then
$$ \chi(\mathfrak A(X_n,Y_m))\ge\frac{mn}{9\alpha(S)\|X_n\|\cdot\|Y_m^*\|}, $$
where $\|X_n\|=\|x_1+\dots+x_n\|$, $\|Y_m^*\|=\|y'_1+\dots+y'_m\|$, and $\chi(E)$ denotes the local unconditional constant of $E$.
Using this theorem we can ascertain the absence of local unconditional structure in some spaces of operators (see theorem 2 and propositions 1–7). In particular $\prod_p(\ell_n,\ell_v)$, $N_p(\ell_u,\ell_v)$ have no local unconditional structure provided $\max(1/2,1/p)<1/u'$ or $\max(1/2,1/p')<1/v'$, $1<u,v,p<\infty$}.