Fractional powers of a class of mellin multiplier transforms part i

Abstract

The paper is concerned with a class of Mellin multiplier transforms, mapping one weighted L^P(0,) space into another, whose symbols are of a particular form. An expression is easily obtained for positive integral powers of such operators and this forms the basis of an extension to fractional powers, A rigorous framework for the analysis is described. Analogues of the index laws of ordinary algebra are established under stated conditions, Connections between powers of an operator and of its adjoint are explored. The theory is illustrated by means of simple integral operators. These examples serve to indicate some of the limitations of a classical setting and provide the stimulus for a distributional treatment which will be presented elsewhere.