Absolutely summing linear operators into spaces with no finite cotype (Q2389179)

A continuous operator between Banach spaces \(X\) and \(Y\) is said to be \((q,p)\)-summing if it sends weakly \(p\)-summable sequences in \(X\) into \(q\)-summable sequences in \(Y\). The relationship between this notion and the cotype of the spaces \(X\) and \(Y\) has been extensively studied. It can be said that the interest on this subject was initiated by the seminal work of Maurey, from which it follows that, if \(X\) (respectively, \(Y\)) has cotype \(q\), then \(\mathcal L(X; Y) = \Pi_{q, 1}(X; Y)\) for all Banach spaces \(Y\) (respectively, \(X\)). In this paper, the authors address the question of whether or not every continuous linear operator in \(\mathcal L(X; Y)\) is absolutely \((q,p)\)-summing for spaces \(Y\) with no finite cotype. In this situation, it is shown that for any infinite-dimensional Banach space \(X\), one has \(\mathcal L(X; Y) \neq \Pi_{p}(X; Y)\) for any \(p \geq 1\), where \(\Pi_{p}(X; Y)\) denotes the \((p,p)\)-summing operators. With an additional hypothesis on \(X\), it is proved that \(\mathcal L(X; Y) \neq \Pi_{q,p}(X; Y)\) for several choices of \(p\) and \(q\). This happens for \(X\) such that \(\ell_r\) is finitely representable in \(X\) whenever \(1\leq q< r\) or \(p>r^*\) (\(r^*\) denotes the conjugate index of \(r\)). Also, considering the index \(r_X\), the infimum of \(q\) such that \(X\) has cotype \(q\), more results can be obtained. The authors prove that \(\mathcal L(X; Y) \neq \Pi_{q,p}(X; Y)\) if \(1\leq q< r_X\) or \(p \geq r_X^*\) or \(1<p<r_X^*\) and \(q< (1/p - 1/r_X^*)^{-1}\). On the other hand, it is shown that \(\mathcal L(X; Y) = \Pi_{q,p}(X; Y)\) for \(p=1\) and \(q>r_X\) or \(1<p<r_X^*\) and \(q >(1/p - 1/r_X^*)^{-1}\). For spaces having cotype \(r_X\), the problem is completely settled in Corollary 2.4. The article ends with some applications to the theory of dominated multilinear mappings.