Operator valued measures as multipliers of L1 (I, X) with order convolution
2015
scientific article
english
- vector valued multiplier
- operator valued Measure
- order convolution
EN Let I = (0, ∞) with the usual topology and product as max multiplication. Then I becomes a locally compact topological semigroup. Let X be a Banach Space. Let L1(I, X) be the Banach space of X-valued measurable functions ƒ such that ʃ0∞ǁƒ (t)ǁdt < ∞. If ƒ ϵ L1(I) and g ϵ L1(I, X), we define ƒ* g(s) = ƒ (s) ʃ0s g(t)dt + g(s) ʃ0s ƒ (t)dt. It turns out that ƒ * g ϵ L1(I, X) and L1(I, X) becomes an L1 (I)-Banach module. A bounded linear operator T on L1 (I, X) is called a multiplier of L1 (I, X) if T(ƒ * g) = ƒ * Tg for all ƒ ϵ L1 (I) and g ϵ L1 (I, X). We characterize the multipliers of L1 (I, X) in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari[12].
41 - 58
CC BY-NC-ND (attribution - noncommercial - no derivatives)
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