The absence of local unconditional structure in some spaces of operators

V. G. Samarskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 300-306

Résumé

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 $\max(1/2,1/p')1/v'$, $1$}.