Order convolution and vector-valued multipliers
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 L¹(I) of all Lebesgue integrable functions on I becomes a commutative semisimple Banach algebra with order convolution as multiplication. A bounded linear operator T on L¹(I) is called a multiplier of L¹(I) if T(f*g) = f*Tg for all f,g ∈ L¹(I). The space of multipliers of L¹(I) was determined by Johnson and Lahr. Let X be a Banach space...