STUDIES IN THE THEORY OF MULTIPLIERS WITH APPLICATIONS TO SEMIGROUPS OF OPERATORS

Abstract

This thesis divides naturally into two parts. The first half is a study of the general theory of multipliers, while the second half deals with applications of the theory of multipliers to the theory of semigroups of operators defined on a Banach spice. We study the multiplier problem for an abstract Hilbert space li, and generalise to H certain important results established for La(G)-multipliers (Larsen [12], Hewitt and Ross [8]). A significant result of this study is the Identification of certain projection operators on L2(G) which are, in several respects. Like the translation operators on Ls (G). We also discuss the restricted multiplier problem for the Banach algebra L1(G) of all absolutely integrable complex-valued functions defined on a compact group G, and we obtain results which are analogous to those obtained by Brainerd and Edwards [1] for L1 (G), where G is a locally compact abelian group. In connection with, semigroups of operators, we discuss, in the context of various Banach spaces, the representation of the multipliers which arise semigroups of operators on these Banach spaces. In this respect, we extend the results of Hille and Phillips [10] proved for the circle group (and generalised to compact abelian groups by Olubummo A. and Babalola V.A. [13]) to certain Banach spaces which are not even function spaces. All these results put together provide a good link between the theory of multipliers for a Banach space and the theory of semigroups of operators on the Banach space.