Spectral measures and automatic continuity

Let X be a locally convex Hausdor space (briey, lcHs) and L(X )d enote 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 -additive map P :! Ls(X); dened on a -algebra of subsets of some set , which is multiplicative (i.e. P (E\ F )= P (E)P (F )f orE;F2 ) and satises P () = 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 :! Ls(X) is, in particular, a vector measure (in the usual sense, [9]) it has an associated spaceL 1 (P )o fP -integrable functions. For each x 2 X; there is an induced X-valued vector measure Px :! X dened by Px: E7! P (E)x; for E2 ; and its associated spaceL 1 (Px )o fPx-integrable functions. It is routine to check that every (C-valued) functionf2L 1 (P ) necessarily belongs toL 1 (Px); for each x2 X; and that the continuous linear operator P (f )= R fdP in X satises P (f)x = R fd(Px); for each x2 X: