Algebras of unbounded scalar-type spectral operators (Q1101661)

The main result of the paper is as follows. Let P:\(\Sigma\to L(X)\) be a closed spectral measure on the quasicomplete locally convex space X and T a densely defined linear operator on X with domain invariant under each operator of the form \(\int_{\Omega}fdP\), where f is a complex bounded \(\Sigma\)-measurable function. Then T is closable and there exists a complex \(\Sigma\)-measurable function g such that the closure of T is the spectral integral \(\int_{\Omega}gdP\) if and only if T leaves invariant each closed subspace of X that is invariant under the range of the spectral measure P. This generalizes work of \textit{A. R. Sourour}, J. Funct. Anal. 29, 16-22 (1978; Zbl 0407.47019) who proved the result above for X a Banach space.