Spectral distributions and generalization of Stone's theorem |
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Authors: | M Balabane H Emamirad M Jazar |
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Institution: | (1) Département de Mathématiques et d'Informatiques, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France;(2) Département de Mathématiques, Université de Poitiers, 86022 Poitiers Cedex, France |
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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. |
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Keywords: | 47B40 47A60 |
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