The canonical spectral measure in Köthe echelon spaces. (English) Zbl 1109.46047

Bonet, J.; Ricker, W. J.

Integral Equations Oper. Theory 53, No. 4, 477-496 (2005).

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.

Reviewer: Heinz Junek (Potsdam)

MSC:

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.

Keywords:

vector measure; density condition; integrable function; Fréchet space