SPECTRAL THEORY AND FUNCTIONAL-CALCULUS FOR OPERATORS ON SPACES OF GENERALIZED-FUNCTIONS

The generalized eigenfunction expansion theory of Zemanian for a differential operator with discrete spectrum is extended to an arbitrary self-adjoint operator on a separable Hilbert space. The extension of the operator to an appropriately defined space of distributions is shown to have a spectral resolution in that space and a functional calculus is developed. Applications to the solution of distributional initial-boundary value problems via semigroup theory are considered. © 1992.