Tenson products of $p$-absolutely summing operators and right ($I_p$, $N_p$) multipliers

E. D. Gluskin   ; S. V. Kislyakov   ; O. I. Reinov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 85-102

Résumé

The article investigates the question of coincidence of some classes of operators, acting from Banach spaces whose duals do not satisfy the $RN$ condition. Separable Banach spaces $JT_r$, $r\in[1,\infty)$, with the following properties are constructed: 1) For each $r$, $r\ge1$, the space $JT_r$ does not contain subspaces isomorphic to $\ell_1$ and has non-separable dual. 2) for each $p$, $p\in(1,\infty)$, and for every Banach space $Z$ $I_p(JT_r,Z)=N_p(JT_r,Z)$. 3) If $1<r<2$ then for each $p$, $p\in(1,r')$, and for every Banach space $Z$ $I_p(JT_r,Z)=N_p(JT_r,Z)$ and for each $p$, $p\ge r'$, there is a $p$-integral operator on $JT_r$ which is not quasi-$p$-nuclear. 4) If $2\le r<\infty$ then for each $p$, $p\ge1$, there is a $p$-integral operator on $JT_r$ which is not quasi-$p$-nuclear. 5) If $1\le r<2$ then $\Pi_1(JT_r,Z)=N_1Q(JT_r,Z)$ for every Banach space $Z$. The above properties of the spaces $JT_r$, are obtained by means of a theorem on tensor products of absolutely $p$-summing operators. This theorem also (as simple corollaries) some recent generalizations of Grothendieck's inequality (see, for example, [7]).