Stephen Montgomery-Smith and Paulette Saab, \(p\)-summing operators on injective tensor products of spaces.
B. Royal Soc. Edin. 120A, (1992), 283-296.
Let \(X\), \(Y\) and \(Z\) be Banach spaces, and let \(\Pi_p(X,Y)\) \((1 \le p < \infty)\) denote the space of \(p\)-summing operators from \(Y\) to \(Z\). We show that, if \(X\) is a \(\mathcal L_\infty\)-space, then a bounded linear operator \(T:X\otimes_\epsilon Y\to Z\) is 1-summing if and only if a naturally associated operator \(T^\#:X\to \Pi_1(Y,Z)\) is 1-summing. This result need not be true if \(X\) is not a \(\mathcal L_\infty\)-space. For \(p>1\), several examples are given with \(X = C[0,1]\) to show that \(T^\#\) can be \(p\)-summing without \(T\) being \(p\)-summing. Indeed, there is an operator \(T\) on \(C[0,1]\otimes_\epsilon \ell_1\) whose associated operator \(T^\#\) is 2-summing, but for all \(N\in\mathbb N\), there exists an \(N\)-dimensional subspace \(U\) of \(C[0,1]\otimes_\epsilon \ell_1\) such that \(T\) restricted to \(U\) is equivalent to the identity operator on \(\ell_\infty^N\). Finally, we show that there is a compact Hausdorff space \(K\) and a bounded linear operator \(T:C(K) \otimes_\epsilon \ell_1 \to \ell_2\) for which \(T^\#:C(K) \to \Pi_1(\ell_1,\ell_2)\) is not 2-summing.
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