Order convolution and vector-valued multipliers

U. B. Tewari. "Order convolution and vector-valued multipliers." Colloquium Mathematicae 108.1 (2007): 53-61. <http://eudml.org/doc/284314>.

@article{U2007,
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 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).},
author = {U. B. Tewari},
journal = {Colloquium Mathematicae},
keywords = {interval; order convolution; semigroup; multiplier; operator-valued function; vector-valued function},
language = {eng},
number = {1},
pages = {53-61},
title = {Order convolution and vector-valued multipliers},
url = {http://eudml.org/doc/284314},
volume = {108},
year = {2007},
}

TY - JOUR
AU - U. B. Tewari
TI - Order convolution and vector-valued multipliers
JO - Colloquium Mathematicae
PY - 2007
VL - 108
IS - 1
SP - 53
EP - 61
AB - 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).
LA - eng
KW - interval; order convolution; semigroup; multiplier; operator-valued function; vector-valued function
UR - http://eudml.org/doc/284314
ER -