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Boolean algebras of projections and ranges of spectral measures. (English) Zbl 0961.46034
Starting from simple examples showing the existence of elementary operators which “ought-to-be” scalar-type spectral operators in the sense of N. Dunford but they are not, whereas the difficulties often disappear when the norm topology is replaced by a weaker one, the authors are led to a systematic study of the spectral measures in the context of the locally convex spaces. Let \(X\) be a locally convex Hausdorff space and let \(L_s(X)\) be the space of all continuous operators on \(X\) equipped with the topology of pointwise convergence. Let also \(\Sigma\) be a \(\sigma\)-algebra of subsets of some nonempty set. The authors are mainly interested in spectral measures \(P:\Sigma\to L_s(X)\). They try to answer whether or not certain standard assumptions are essential. For instance, when the fact that \(X\) complete (or quasicomplete, or sequentially complete) is important and when not? Or when the equicontinuity (or the weak compactness) of the range of a spectral measure is important and when not? There are many suitable examples and pertinent open questions related to the basic phenomena.

46G10 Vector-valued measures and integration
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.