Banach function spaces and spectral measures

• Main Author: Byrne, Catriona M.
• Published: University of Edinburgh 1982
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.642282

The fundamental link between prespectral measures and Banach function spaces is to be found in a theorem of T.A. Gillespie which relates cyclic spaces isomorphically to certain Banach function spaces. We obtain here an extension of this result to the wider class of precyclic spaces. We then consider the properties of weak sequential completeness and reflexivity in Banach function spaces: necessary and sufficient conditions are obtained which in turn, via the aforementioned isomorphisms, both extend and simplify analogously formulated existing results for cyclic spaces. Finally the concept of a homomorphism between pairs of Banach function spaces is examined. The class of such mappings is determined and a complete description obtained in the form of a (unique) disjoint sum of two mappings, one of which is always an isomorphism and the other of which is arbitrary in a certain sense, or null. It is shown moreover that the isomorphic component itself is composed of two other isomorphisms in a manner analogous to the geometrical composition of a rotation and a dilatation.