Closed spectral measures in Fréchet spaces (Q1066056)

This note presents sufficient conditions ensuring the closedness of certain operator-valued measures (i.e. their \(L^ 1\)-space is complete). It is shown that if X is a separable Fréchet space, then any L(X)- valued operator measure is closed; here L(X) is the space of all continuous linear operators in X equipped with the strong operator topology. For an L(X)-valued spectral measure P, where X is an arbitrary Fréchet space, it is shown that if P is interpreted as the projective limit of a suitable sequence of Banach space-valued spectral measures \(\{P_ n\}\), induced by P, then P is closed if and only if each \(P_ n\) is closed, thereby reducing the question of closedness for P to the same, but more tractable, problem in Banach spaces.