ISSN:
1572-9036
Keywords:
47B40
;
47A60
;
Spectral distribution
;
scalar-type spectral operators
;
Stone's theorem
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In this paper, we introduce the notion ofspectral distribution which is a generalization of the spectral measure. This notion is closely related to distribution semigroups and generalized scalar operators. The associated operator (called themomentum of the spectral distribution) has a functional calculus defined for infinitely differentiable functions on the real line. Our main result says thatA generating a smooth distribution group of orderk is equivalent to having ak-times integrated group that are O(¦t¦ k ) oriA being the momentum of a spectral distribution of degreek. We obtain the standard version of Stone's theorem as a special case of this result. The standard properties of a functional calculus together with spectral mapping theorem are derived. Finally, we show how the degree of a spectral distribution is related to the degree of the nilpotent operators which separate its momentum from its scalar part.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00997121
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