Multilinear variants of Pietsch's composition theorem
Let \(1<p,q<\infty\) and \(1 \leq r <\infty\) be such that \(1/r=1/p+1/q\). The composition theorem for absolutely summing linear operators [\textit{A.\ Pietsch}, Stud.\ Math.\ 28, 333--353 (1967; Zbl 0156.37903)] then states that \(\Pi_q \circ \Pi_p \subset \Pi_r\). Various concepts of multilinear summing operators were introduced in the past years, by extending the well-known one from the linear case. In this paper, the author proves that, as in the linear case, there is a splitting theorem for dominated operators (with a nontrivial proof of more than four pages). As a consequence of this result, he proves various multilinear variants of Pietsch's composition theorem. More precisely, let \(\delta_p^n\) (resp., \(\Pi_r^n\)) denote the class of \(p\)-dominated (resp., absolutely \(r\)-summing) \(n\)-linear operators. Then it is first shown that one of the possible extensions does not hold true: \(\Pi_q \circ \delta_p^n \subset \delta_r^n\) if and only if \(n=1\). Second, several consequences of the splitting theorem eventually lead to the following positive statement: \(\Pi_q \circ \delta_p^n \subset \Pi_r^n\), provided that \(1 \leq r \leq 2\). The case \(2<r<\infty\) is left open as a challenging task.
scientific article; zbMATH DE number 5757965
scientific article