NON-WEAK COMPACTNESS OF THE INTEGRATION MAP FOR VECTOR MEASURES

Let m be a vector measure with values in a Banach space X . If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map I(m) which sends each f of L1(m) to the element integral f dm (of X) . Many properties of the (continuous) operator I(m) are closely related to the nature of the space L1(m) . In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that I(m) is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1([-pi, pi]) , the Volterra operator on L1([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps I(m) (or their restrictions) for suitable measures m.