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 dev eloped. Spectral theory and a functional calculus are shown to be vali d in these spaces. Application of the theory is illustrated by conside ring a normal realisation of the operator x(d/dx) and demonstrating th at 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 operator s arise as functions of x(d/dx) and so a distributional theory of thes e operators can be obtained using the functional calculus in the gener alized function space. (C) 1996 Academic Press, Inc.