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 f * g(s) = ƒ (s) ʃ g(t)dt + g(s) ʃ0s ƒ (t)dt. It turns out that f * 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(f * g) = f * 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].
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