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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References
Arendt, W.: Resolvent positive operators,Proc. London Math. Soc. (3)54 (1987), 321–349.
Arendt, W.: Vector valued Laplace transforms and Cauchy problems,Israel J. Math. (3)59 (1987), 327–352.
Arendt, W.: Sobolev imbeddings and integrated semigroups, in P. Clémentet al. (eds),Trends in Semigroup Theory and Evolution Equations, Marcel Dekker, New York, 1991, pp. 29–40.
Arendt, W. and Kellermann, H.: Integrated solutions of Volterra integro-differential equations and applications, Semester-bericht Functionalanalysis, Tübingen, Sommersemester, 1987.
Balabane, M.: Quelques propositions pour un calcul symbolique, Thése d'Etat, Université de Paris 7, 1982.
Balabane, M. and Emamirad, H.: Smooth distribution group and Schrödinger equation inL p(ℝn),J. Math. Anal. Appl. 70 (1979), 61–71.
Balabane, M. and Emamirad, H.: Pseudo-Differential Parabolic Systems in Lp(ℝn);Contributions to Non-linear P.D.E., Research Notes in Mathematics, vol. 89, Pitman, New York, 1983.
Balabane, M. and Emamirad, H.:L p estimates for Schrödinger evolution equations,Trans. Amer. Math. Soc. (1) 292 (1985), 357–373.
Colojoará, I. and Foiaş, C.:Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968.
Davies, E. B.:One-parameter Semigroups, Academic Press, New York, 1980.
deLaubenfels, R.: Integrated semigroups,C-semigroups and the abstract Cauchy problem,Semigroup Forum 41 (1990), 83–95.
deLaubenfels, R.: Polynomials of generator of integrated semigroups,Proc. Amer. Math. Soc. (1) 107 (September 1989).
deLaubenfels, R.: Integrated semigroups and integrodifferential equations,Math. Z. 204 (1990), 501–514.
Dunford, N.:Spectral Theory in Topological Vector Spaces, Functions, Series, Operators, Vols. I, II, Colloq. Math. Soc. János Bolyai, 35, North-Holland, Amsterdam, New York, 1983, pp. 391–422.
Dunford, N. and Schwarte, J.:Linear Operators III, Spectral Operators, Wiley-Interscience, New York, 1971.
Emamirad, H. and Jazar, M.: Applications of spectral distributions to some Cauchy problems inL Pℝn), in P. Clémentet al. (eds),Trends in Semigroup Theory and Evolution Equations, Marcel Dekker, New York, 1991, pp. 143–151.
Foiaş, C.: Une application des distributions vectorielles à le théories spectrale,Bull. Sci. Math. (2)84 (1960), 147–158.
Hejtmanek, J.: Dynamics and spectrum of the linear multiple scattering operator in the Banach lattice L1 (ℝ3xℝ3),Transport Theory Statist. Phys. 8 (1979), 29–44.
Jazar, M.: Sur la théorie de la distribution spectrale et applications aux problèmes de Cauchy, Thèse de l'Université de Poitiers, 1991.
Kellermann, H. and Hieber, M.: Integrated semigroups,J. Funct. Anal. 84 (1989), 160–180.
Kritt, B.: A theory of unbounded generalized scalar operator,Proc. Amer. Math. Soc. (2)32 (1972), 484–490.
Kritt, B.: The Fourier transform of an unbounded spectral distributions,Proc. Amer. Math. Soc. 35 (1972), 74–80.
Neubrander, F.: Integrated semigroups and their applications to the abstract Cauchy problem,Pacific J. Math. 135 (1988), 111–155.
Ricker, W. J.: Functional calculi for the Laplace operator inL p(ℝ), Miniconference on Harmonic Analysis, Canberra, Australia, June 1987,Proc. Centre Math. Anal. 15 (1987), 242–254.
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Balabane, M., Emamirad, H. & Jazar, M. Spectral distributions and generalization of Stone's theorem. Acta Appl Math 31, 275–295 (1993). https://doi.org/10.1007/BF00997121
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DOI: https://doi.org/10.1007/BF00997121