Operational Calculus

Abstract

The Operational Calculus is a construction used for analyzing the behavior of linear operators that arise in the study of ordinary and partial differential equations. Given a linear operator T and a class of functions F, one rigorously defines a new operator f(T) for each f in F and establishes properties of the transformation f -> f(T), among which is that, if F is an algebra of functions, then the transformation induces an algebra homomorphism from F to the algebra of bounded linear operators on a Banach space. This paper begins with a discussion of an operational calculus for compact symmetric operators. This motivates the construction of the Dunford operational calculus for general bounded linear operators. Next, a treatment for bounded symmetric operators is provided, together with a rigorous presentation of all background material. All this is the basis of an operational calculus for unbounded symmetric operators T on a complex Hilbert space. This latter construction is based on a representation theorem of Riesz and Lorch for unbounded self-adjoint operators: the presentation is simpler and more illuminating than the customary one.