Spectral Measures In Abstract Spaces

We study the theory of spectral measures in topological vector spaces. We extend the Hilbert space theory to this setting and generalize the notion of spectral measures in some useful ways to provide a framework for operator theory in this setting. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be “spectral”. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon-Nikodym theorem is proved. We then extend the theory of spectral measures to the case where values are assumed in the set of discontinuous (in normed spaces ”un-bounded”) operators. Examples of operators in nonlocally convex spaces are given which have densely defined measures. AMS(MOS) subject classifications. Primary 47B40; Secondary 28B05, 46A15. 46H05.