zbMATH — the first resource for mathematics

The canonical spectral measure in Köthe echelon spaces. (English) Zbl 1109.46047
Spectral measures on Banach spaces are well understood. In contrast to this, new phenomena occur in the locally convex setting. In the present paper, a detailed study of the properties of the canonical spectral measure \(P\) on a Köthe echelon sequence space \(\Lambda\) is presented. (The canonical spectral measure is defined by \(P(E)(x)= \sum_{j \in E}\langle x,e_j\rangle e_j\) for \(E \subseteq \mathbb N\) and \(x \in \Lambda\).) It turns out that there are fundamental connections between the topological and geometric structure of \(\Lambda\) (e.g., Schwartz, nuclear, Montel, satisfying the density condition, etc.) on the one hand, and operator- and measure theoretic properties of \(P\) (e.g., compact range, boundedly \(\sigma\)-additivity, finite variation) on the other hand. In a final section, the set \({\mathcal L}^1(P)\) of \(P\)-integrable functions is characterized in terms of multipliers of \(\Lambda\), and it is shown that \({\mathcal L}^1(P)={\mathcal L}^\infty(P)\) holds true iff no sectional subspace of \(\Lambda\) is Schwartz.

46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
46A45 Sequence spaces (including Köthe sequence spaces)
46A04 Locally convex Fréchet spaces and (DF)-spaces
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
Full Text: DOI