Vector Daniell integrals
Compactness properties of the integration mapassociated with a vector measure
Semivariation in -spaces
Suppose that and are Banach spaces and that the Banach space is their complete tensor product with respect to some tensor product topology . A uniformly bounded -valued function need not be integrable in with respect to a -valued measure, unless, say, and are Hilbert spaces and is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index and suppose that and are -spaces with the associated -tensor product...
Criteria for weak compactness of vector-valued integration maps
Criteria are given for determining the weak compactness, or otherwise, of the integration map associated with a vector measure. For instance, the space of integrable functions of a weakly compact integration map is necessarily normable for the mean convergence topology. Results are presented which relate weak compactness of the integration map with the property of being a bicontinuous isomorphism onto its range. Finally, a detailed description is given of the compactness properties for the integration...
Operator ideal properties of vector measures with finite variation
Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...
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