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Fractional powers of momentum of a spectral distribution


Author: M. Jazar
Journal: Proc. Amer. Math. Soc. 123 (1995), 1805-1813
MSC: Primary 47D03; Secondary 35J10, 35P05, 47A60, 47N20
DOI: https://doi.org/10.1090/S0002-9939-1995-1242090-0
MathSciNet review: 1242090
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Abstract: In this paper we construct fractional and imaginary powers for the positive momentum B of a spectral distribution and prove the basic properties.

The main result is that for any $ \alpha > 0, - {B^\alpha }$ generates a bounded strongly continuous holomorphic semigroup of angle $ \frac{\pi }{2}$. In particular for $ \alpha = 1$, using Stone's generalized theorem, if iB generates a k-times integrated group of type $ O(\vert t{\vert^k})$ with $ \sigma (B) \subset [0, + \infty [$, then -B generates a strongly continuous holomorphic semigroup of angle $ \frac{\pi }{2}$. A similar corollary is given in the regularized group situation.


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  • [1] W. Arendt and H. Kellermann, Integrated solutions of Volterra integro-differential equations and applications, Semester-bericht Functionalanalysis, Tübingen, Sommersemester, 1987.
  • [2] M. Balabane, Puissances fractionnaires d'un opérateur d'un semi-groupe distribution régulier, Ann. Inst. Fourier (Grenoble) 26 (1976), 157-203. MR 0402534 (53:6353)
  • [3] M. Balabane and H. Emamirad, Smooth distribution group and Schrödinger equation in $ {L^p}({\mathbb{R}^n})$, J. Math. Anal. Appl. 70 (1979), 61-71. MR 541059 (80j:46063)
  • [4] -, Pseudo-differential parabolic systems in $ {L^p}({\mathbb{R}^n})$, Contributions to Non-Linear P. D. E., Pitman Res. Notes Math. Ser., vol. 89, Longman Sci. Tech., Harlow, 1983.
  • [5] M. Balabane, H. Emamirad, and M. Jazar, Spectral distributions and generalization of Stone's Theorem to the Banach space, Acta Appl. Math. 31 (1993), 275-295. MR 1232940 (94f:47038)
  • [6] Balakrishnan, Fractional powers of closed operators, Pacific J. Math. 10 (1960).
  • [7] S. Bochner, Diffusion equations and stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 369-370. MR 0030151 (10:720i)
  • [8] G. Da Prato, Semigruppi regolarizzabili, Richerche Mat. 15 (1966), 233-248. MR 0225199 (37:793)
  • [9] R. deLaubenfels, Integrated semigroups, C-semigroups and the abstract Cauchy problem, Semigroup Forum 41 (1990), 83-95. MR 1048324 (91b:47092)
  • [10] -, C-semigroups and the abstract Cauchy problem, J. Funct. Anal. 111 (1993), 44-61.
  • [11] -, Totally accretive operators, Proc. Amer. Math. Soc. 103 (1988), 551-556. MR 943083 (89g:47044)
  • [12] R. deLaubenfels and K. Boyadzhiev, Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc. 118 (1993), 113-118. MR 1128725 (93f:47043)
  • [13] R. deLaubenfels, H. Emamirad, and M. Jazar, Regularized scalar operators (submitted).
  • [14] H. Emamirad and M. Jazar, Application of spectral distributions to some Cauchy problems in $ {L^p}({\mathbb{R}^n})$, Trends in Semigroup Theory and Evolution Equations (P. Clément, ed.), Marcel Dekker, New York, 1991, pp. 143-151. MR 1164647 (93h:47048)
  • [15] M. Hieber, Laplace transforms and $ \alpha $-times integrated semigroups, Forum Math. 3 (1991), 595-612. MR 1130001 (92k:47075)
  • [16] M. Jazar, Sur la théorie de la distribution spectrale et applications aux problèmes de Cauchy, Thèse de l'Université de Poitiers, 1991.
  • [17] O. E. Lanford III and D. W. Robinson, Fractional powers of generators of equicontinuous semigroups and fractional derivatives, J. Austral. Math. Soc. Ser. A 46 (1989), 437-504. MR 987564 (90d:47040)
  • [18] C. Martinez, M. Sanz, and L. Marco, Fractional powers of operators, J. Math. Soc. Japan (2) 40 (1988). MR 930604 (89c:47021)
  • [19] I. Miyadera and N. Tanaka, Some remarks on C-semigroups and integrated semigroups, Proc. Japan Acad. Ser. A 63 (1987). MR 906977 (89i:47077)
  • [20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983. MR 710486 (85g:47061)
  • [21] K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1964.

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DOI: https://doi.org/10.1090/S0002-9939-1995-1242090-0
Article copyright: © Copyright 1995 American Mathematical Society

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