Spectral measures and automatic continuity
نویسندگان
چکیده
Let X be a locally convex Hausdorff space (briefly, lcHs) and L(X) denote the space of all continuous linear operators of X into itself. The space L(X) is denoted by Ls(X) when it is equipped with the topology of pointwise convergence in X (i.e. the strong operator topology). By a spectral measure in X is meant a σ-additive map P : Σ → Ls(X), defined on a σ-algebra Σ of subsets of some set Ω, which is multiplicative (i.e. P (E ∩ F ) = P (E)P (F ) for E,F ∈ Σ) and satisfies P (Ω) = I, the identity operator in X. This concept is a natural extension to Banach and more general lc-spacesX of the notion of the resolution of the identity for normal operators in Hilbert spaces, [7,11,19,21]. Since a spectral measure P : Σ → Ls(X) is, in particular, a vector measure (in the usual sense, [9]) it has an associated space L(P ) of P -integrable functions. For each x ∈ X, there is an induced X-valued vector measure Px : Σ → X defined by Px : E 7→ P (E)x, for E ∈ Σ, and its associated space L(Px) of Px-integrable functions. It is routine to check that every (C-valued) function f ∈ L(P ) necessarily belongs to L(Px), for each x ∈ X, and that the continuous linear operator P (f) = ∫ Ω fdP in X satisfies P (f)x = ∫ Ω fd(Px), for each x ∈ X. The topic of this note is the converse question. Namely, suppose that f is a C-valued, Σ-measurable function which belongs to L(Px), for each x ∈ X. Then the map P[f ] : X → X defined by
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