A Spectral Approach to Distributional Extensions of Normal Operators

A constructive method for producing a test function space and hence a generalized function space suitable for a given normal operator is developed. Spectral theory and a functional calculus are shown to be valid in these spaces. Application of the theory is illustrated by considering a normal realisation of the operator x drdx.and demonstrating that the test and generalized function spaces for this operator coincide with certain spaces originally introduced by McBride in the theory of fractional calculus. It is shown that the so-called Hadamard operators arise as functions of x drdx.and so a distributional theory of these operators can be obtained using the functional calculus in the generalized function space