Weak compactness of the integration map associated with a spectral measure (Q1344165)

The author shows that if \(P\) is a closed, equicontinuous spectral measure in a l.c. space \(X\), then its associated integration map \(I_p\) is weakly compact if and only if \(P\) has a finite range. The result is proved with use of technique of vector measure integration theory and locally convex algebras. The equicontinuity condition can be replaced by countable additivity of \(P\) in uniform convergence topology on bounded sets in \(X\).