Order convolution and vector-valued multipliers

Abstract

Let I = (0, ∞) with the usual topology. For x, y ∈ I, we define xy = max(x, y). Then I becomes a locally compact commutative topological semigroup. The Banach space L 1 (I) of all Lebesgue integrable functions on I becomes a commutative semisimple Banach algebra with order convolution as multiplication. A bounded linear operatorThe space of multipliers of L 1 (I) was determined by Johnson and Lahr. Let X be a Banach space and L 1 (I, X) be the Banach space of all X-valued Bochner integrable functions on I. We show that L 1 (I, X) becomes an L 1 (I)-Banach module. Suppose X and Y are Banach spaces. A bounded linear operator T from L 1 (I, X) to L 1 (I, Y ) is called a multiplier if T (f ⋆ g) = f ⋆ T g for all f ∈ L 1 (I) and g ∈ L 1 (I, X). In this paper, we characterize the multipliers from L 1 (I, X) to L 1 (I, Y ).

Operator Valued Measures as Multipliers of L1(I,X) with order Convolution

“…Similar to Larsen's approach [11], we characterize multipliers on L 1 (I) in terms of absolutely continuous functions onÎ. Tewari [12] had also noted this. If T is a multiplier of L 1 (I) then there exists a unique µ in M(I) of the form µ = αδ + h, α ∈ C, h ∈ L 1 (I) such that T f = µ * f ∀f ∈ L 1 (I).…”

“…The following proposition tells us that {u n } acts as an approximate identity for L 1 (I, X) (see Proposition 3.1, [12]). The following proposition follows immediately from Proposition 3.2, [12].…”

“…In Section 5, using Larsen's [11] ideas and the technique of Tewari, Dutta and Vaidya [13], we characterize the multipliers of L 1 (I) to L 1 (I, X) and then multipliers of L 1 (I, X) to L 1 (I, Y ). We characterize these multipliers in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari [12].…”

“…Tewari [12] had characterized these multipliers in terms of operator valued functions. In this section, using Larsen's [11] ideas and the technique of Tewari,Dutta and Vaidya [13], we characterize the multipliers of L 1 (I) to L 1 (I, X) and then multipliers of L 1 (I, X) to L 1 (I, X) in terms of operator valued measures with point-wise finite variation.…”

“…A bounded linear operator T on L 1 (I, X) is called a multiplier of L 1 (I, X) if T (f * g) = f * T g for all f ∈ L 1 (I) and g ∈ L 1 (I, X). We characterize the multipliers of L 1 (I, X) in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari [12].…”