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 and L¹(I,X) be the Banach space of all X-valued Bochner integrable functions on I. We show that L¹(I,X) becomes an L¹(I)-Banach module. Suppose X and Y are Banach spaces. A bounded linear operator T from L¹(I,X) to L¹(I,Y) is called a multiplier if T(f*g) = f*Tg for all f ∈ L¹(I) and g ∈ L¹(I,X). In this paper, we characterize the multipliers from L¹(I,X) to L¹(I,Y).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-5,
author = {U. B. Tewari},
title = {Order convolution and vector-valued multipliers},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {53-61},
zbl = {1113.43003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-5}
}
U. B. Tewari. Order convolution and vector-valued multipliers. Colloquium Mathematicae, Tome 107 (2007) pp. 53-61. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm108-1-5/