Fully summing multilinear and holomorphic mappings into Hilbert spaces

It is known that whenever $E_{1}$, \dots, $E_{n}$ are infinite dimensional $\mathcal{L}_{\infty}$-spaces and $F$ is any infinite dimensional Banach space, there exists a bounded $n$-linear mapping that fails to be absolutely $(1;2)$-summing. In this paper we obtain a sufficient condition in order to assure that a given $n$-linear mapping $T$ from infinite dimensional $\mathcal{L}_{\infty}$-spaces into an infinite dimensional Hilbert space is absolutely $(1;2)$-summing. Besides, we also give a sufficient condition in order to obtain a fully $(1;1)$-summing multilinear mapping from $l_{1}\times \ldots \times l_{1}\times l_{2}$ into an infinite dimensional Hilbert space. In the last section we introduce the concept of fully summing holomorphic mappings and give the first examples of this kind of maps.