The canonical spectral measure and köothe function spaces

Abstract

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 σ-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.