On the Structure of L1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L-1(m). In this paper we study the cases of the integration operator being: (i) p-concave on L p( m), or (ii) positive p-summing on L-1(m) (where 1 <= p < infinity). We prove that (i) is equivalent to saying that L-1(m) contains continuously the L p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L-1(m) is order isomorphic to the L-1 space of a non-negative scalar measure.