Non-weak compactness of the integration map for vector measures
Journal of the Australian Mathematical Society
Let m be a vector measure with values in a Banach space X . If L l (m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L l (m) is a Banach space. A natural operator associated with m is its integration map I m which sends each / of L l (m) to the element ffdm (of X). Many properties of the (continuous) operator I m are closely related to the nature of the space L (m). In general, it is difficult to identify L (m). We aim to exhibit
... exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L (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 L l ([-n, n\), the Volterra operator on l) ([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 . 1991 Mathematics subject classification [Amer. Math. Soc.): 28B05, 47B05, 45P05. order completeness [1, 5] ) it is also to be expected that other Banach space properties are exhibited which are not typical of the classical situation (see Sections 1 and 2). Our particular interest is in the integration map I m : L l (m) -> X defined by I m f = J fdm, for every / e L l (m). Just as for scalar measures it turns out that I m is always bounded and linear. So, if X is reflexive, then I m is necessarily weakly compact. If we admit non-reflexive spaces X it can happen that I m is weakly compact for some measures m and not weakly compact for others. In this article we wish to concentrate on exhibiting non-trivial (and, hopefully, interesting) examples of non-weakly compact integration maps I m . The case of maps I m which are weakly compact (or even compact) will be taken up elsewhere. The authors wish to thank Ben de Pagter for some useful discussions on this topic.