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On subordinated holomorphic semigroups


Authors: Alfred S. Carasso and Tosio Kato
Journal: Trans. Amer. Math. Soc. 327 (1991), 867-878
MSC: Primary 47D03; Secondary 60J35
DOI: https://doi.org/10.1090/S0002-9947-1991-1018572-4
MathSciNet review: 1018572
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Abstract: If $ [{e^{ - tA}}]$ is a uniformly bounded $ {C_0}$ semigroup on a complex Banach space $ X$, then $ - {A^\alpha },$, $ 0 < \alpha < 1$, generates a holomorphic semigroup on $ X$, and $ [{e^{ - t{A^\alpha }}}]$ is subordinated to $ [{e^{ - tA}}]$ through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property.


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  • [1] A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups, Trans. Amer. Math. Soc. 91 (1959), 330-353. MR 0107179 (21:5904)
  • [2] -, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419-437. MR 0115096 (22:5899)
  • [3] S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 368-370. MR 0030151 (10:720i)
  • [4] -, Harmonic analysis and the theory of probability, Univ. of California Press, Berkeley, Calif., 1955. MR 0072370 (17:273d)
  • [5] W. Feller, An introduction to probability theory and its applications, vol. 2, 2nd ed., Wiley, New York, 1971.
  • [6] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Providence, R.I., 1957. MR 0089373 (19:664d)
  • [7] E. Nelson, A functional calculus using singular Laplace integrals, Trans. Amer. Math. Soc. 88 (1958), 400-413. MR 0096136 (20:2631)
  • [8] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [9] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [10] R. S. Phillips, On the generation of semigroups of linear operators, Pacific J. Math. 2 (1952), 343-369. MR 0050797 (14:383g)
  • [11] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 0210528 (35:1420)
  • [12] K. Yosida, An abstract analyticity in time for solutions of a diffusion equation, Proc. Japan Acad. 35 (1959), 109-113. MR 0105559 (21:4298)
  • [13] -, Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them, Proc. Japan Acad. 36 (1960), 86-89. MR 0121665 (22:12399)
  • [14] -, On a class of infinitesimal generators and the integration problem of evolution equations, Proc. Fourth Berkeley Sympos. Math. Stat. Probab. Volume 2, Contributions to Probability Theory, (Jerzy Neyman, Ed.), Univ. of California Press, Berkeley, Calif., 1961, pp. 623-633. MR 0137009 (25:468)
  • [15] -, On holomorphic Markov processes, Proc. Japan Acad. 42 (1966), 313-317. MR 0207048 (34:6864)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1018572-4
Keywords: Subordinated semigroups, holomorphic extensions
Article copyright: © Copyright 1991 American Mathematical Society

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