Algebras of unbounded scalar-type spectral operators

Abstract

If P: -* Jf (X) is a closed spectral measure in the quasicomplete locally convex space X and if T is a densely defined linear operator in X with domain invariant under each operator of the form / fdP, with / a complex bounded -measurable function then T is closable and there exists a complex -measurable function / such that the closure of T is the spectral integral / fdP if and only if T leaves invariant each closed subspace of X which is invariant under the range of the spectral measure P. 0. Introduction. Let X be a complex locally convex Hausdorff space, assumed quasicomplete throughout. Let <?( X) be the space of all continuous linear operators in X equipped with the topology of pointwise convergence in X. Let P: -> JS?(X) be a spectral measure with domain a -algebra of subsets of some point set and with range a closed subset of JSP(X). The intention of the present paper is to characterize those densely defined operators T on X that can be represented as a spectral integral f^fdP for some complex, -measurable function / on . More precisely, we show (Theorem 6.2) that if T is densely defined with domain 3){T) invariant under each operator of the form J a fdP with / a bounded, complex -measurable function, then T is closable and there exists a complex -measurable function / on such that T is given by the spectral integral J Q fdP if and only if T leaves invariant each closed linear subspace of <&(X) which is invariant under the range of the spectral measure P. That part of the conclusion which asserts that T is closable appears to be new, even if X is a Banach space, and is not valid if the stated assumption on 3){T) is replaced by the weaker assumption that S>(T) is invariant under merely the range of P. Under this weaker assumption on @(T), the above characterization remains valid if, in addition, T is assumed closed (Corollary 6.3) and this result extends to the locally convex setting a characterization of scalar-type spectral operators given by Sourour [22] for the case that X is Banach. Further, for everywhere defined operators on locally convex space, our results specialize to the reflexivity criteria of [6], [4]. 41