Stieltjes integration, spectral analysis and the locally-convex algebra (π΅π)
Gregers L. Krabbe Β· Bulletin of the American Mathematical Society Β· 1965
The space (BV) (of all functions of bounded variation on an interval [b 0 , bi]) is an algebra under pointwise multiplication; one of our aims is to show how it can be made into a locally-convex algebra on which all continuous linear multiplicative functionals are represented by point measures on the interval (b Ql b\].Our main result deals with spectral and non spectral operators.The algebra structure is disregarded in Β§5, where (BV) is endowed with a topology such that the most general continuous linear functional on (BV) has a natural representation by means of a Stieltjes integral.Given a complete barreled space 3Γ, we introduce a family Si of functions whose values are commuting projection operators in 3Β£.The algebra (BV) is topologized in such a way that each strongly-continuous representation g->u(g) (of (BV) on 3Β£) can be expressed in a natural way in terms of some .FGSi; in fact, u(g) is the Stieltjes integral of g with respect to F.