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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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