Weak eigenvectors and the functional calculus

Abstract

In this paper we are concerned with the problem of constructing the functional calculus for an operator with continuous spectrum from a given collection of weak eigenvectors. A functional calculus is a homomorphism f-+ Tf of an algebra of functions defined over the 'spectrum' V into a subalgebra of bounded linear operators acting in a banach space E; we may regard an 'eigenfunction expansion' for such a functional calculus as an expression for all Tf over a banach space D dense in E of the form Tf = fvf(A)TA dm(A), where m is a measure over V and TA is a bounded linear operator from D into the space of 'distributions' D'. In general, it is hoped that the range of T, ImTz, will consist of rather precisely specified weak eigenvectors (specified, for example, by boundary values). Since a family of weak eigenvectors is often available, a problem that often appears in applications is the following: Given a family of weak eigenvectors at A, when is it true that thisfamily coincides with ImT? There are many results in the literature for a closed operator with dense domain in a banach space with continuous spectrum which guarantee a rich functional calculus and a representation of the functional calculus in terms of an eigenfunction expansion where, in fact, ImTz consists of weak eigenvectors. F. Browder [3] established such results for a large class of nonsymmetric partial differential operators, and Gel'fand and Silov [5] were the first to consider the problem of the existence of eigenfunction expansions in general operator theoretic terms. However there are few results that tell us which weak eigenvectors to choose for the eigenfunction expansion. Most results of this nature are confined to differential operators with strong regularity theorems, the most classical being Weyl's construction of the Plancherel formula for a singular second order ordinary differential operator [10]. The crucial step is the 'Weyl lemma' which asserts that certain weak eigenvectors are actually infinitely differentiable, and so, classical uniqueness theorems may be invoked to determine precisely which weak eigenvectors contribute to the eigenfunction expansion. The results given here, in contrast, are well within the context of functional