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The canonical spectral measure and Köthe function spaces. (English) Zbl 1137.46027
Summary: The theme of this paper is the interaction between analytic properties of (Fréchet) Köthe function spaces \(X\) and measure/operator theoretic properties of the canonical spectral measure \(Q\) acting in \(X\). For instance, \(Q\) is boundedly \(\sigma\)-additive iff \(X\) is Montel. Or, \(Q\) has finite variation (for the strong operator topology) iff \(X\) is an AL-space. Or, there exist unbounded \(Q\)-integrable functions whenever \(X\) is non-normable and has the density condition; this is based on characterizing \(Q\)-integrable functions as measurable multipliers.

MSC:
46G10 Vector-valued measures and integration
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B60 Linear operators on ordered spaces
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