A Vector-Valued Operational Calculus and Abstract Cauchy Problems.

Abstract

Initial and boundary value problems for linear differential and integro-differential equations are at the heart of mathematical analysis. About 100 years ago, Oliver Heaviside promoted a set of formal, algebraic rules which allow a complete analysis of a large class of such problems. Although Heaviside's operational calculus was entirely heuristic in nature, it almost always led to correct results. This encouraged many mathematicians to search for a solid mathematical foundation for Heaviside's method, resulting in two competing mathematical theories: (a) Laplace transform theory for functions, distributions and other generalized functions, (b) J. Mikusinski's field of convolution quotients of continuous functions. In this dissertation we will investigate a unifying approach to Heaviside's operational calculus which allows us to extend the method to vector-valued functions. The main components are (a) a new approach to generalized functions, considering them not primarily as functionals on a space of test functions or as convolution quotients in Mikusinski's quotient field, but as limits of continuous functions in appropriate norms, and (b) an asymptotic extension of the classical Laplace transform allowing the transform of functions and generalized functions of arbitrary growth at infinity. The mathematics are based on a careful analysis of the convolution transform $f \to k \star f.$ This is done via a new inversion formula for the Laplace transform, which enables us to extend Titchmarsh's injectivity theorem and Foias' dense range theorem for the convolution transform to Banach space valued functions. The abstract results are applied to abstract Cauchy problems. We indicate the manner in which the operational methods can be employed to obtain existence and uniqueness results for initial value problems for differential equations in Banach spaces.