Inclusion theorems for absolutely summing holomorphic mappings

Abstract:

For linear operators, if $1\leq p\leq q<\infty ,$ then every absolutely $p$-summing operator is also absolutely $q$-summing. On the other hand, it is well known that for $n\geq 2,$ there are no general “inclusion theorems”for absolutely summing $n$-linear mappings or $n$-homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely $p$-summing and absolutely $q$-summing linear operators coincide, and prove that for $1\leq p\leq q\leq 2$ and $n\geq 2$, we have inclusion theorems for absolutely summing $n$-linear mappings/$n$-homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as $p$ increases, the classes of absolutely $p$-summing mappings becomes smaller.