1994

Ph.D

University of Madras

This thesis deals with the study of the multipliers. The problem of characterizing the multipliers from a Segal Algebra S(G) into the space ((L^p)(G))and from S(G) into ((A^p)(G)). A vector version of characterizations of the multipliers for the pair ( (L^1)(G), (L^p)(G) ) is also obtained. Segal Algebras are very important subalgebras of((L^1)(G)). The class of functions introduced by Wiener in 1932 in his study of Tauberian theorems is the very first example of a Segal Algebra. In this thesis, Segal Algebra, Multipliers on Segal Algebra are definded; Many lemmas and Theorems are described, proved with some remarks, and used for discussions of the present study. On discussions over 'Multipliers and A^p(G) algebras, a concrete dual space characterization for the space M(S(G), A^p(G)) where S(G) is a Segal Algebra contained in A^p(G), is obtained. And proved that for 1<p<infinity, M(S(G), A^p(G)) could be identified with a Banach Space of continuous functions. Existence of Isometric Isomorphism of M(S(G),A^p(G)) onto the dual space of a Banach space of a continuous functions is stated in a theorem and proved that on a unit sphere of M(S(G), A^p(G)) the strong operator topology is stronger than the weak*topology. A vector version of the characterizations of the multipliers for the pair (L^(G), L^p(G)), 1<p<infinity, is provided, where G is a locally compact abelian group under the assumption that A is commutative Banach Algebra with a bounded approximate identity. The main theorem, Let T: L^1(G) --> L^p(G,A) be a continuous -linear operator where 1<p<infinity, with some conditions stated to be equivalent and proved.