Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.