J Austral Math Soc Ser A 54 pp287--303, 1993.

Non-Weak Compactness of the Integration Map for Vector Measures

S. Okada and W. J. Ricker

Abstract

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 òf 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-reflective) 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([-p, p]), 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.

1991 AMS Subject Classification: 28B05, 47B05, 45P05

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Authors

S. Okada

University of Tasmania, Hobart, Tas., 7001, Australia.

W. J. Ricker

University of New South Wales, Kensington, NSW, 2033, Australia.