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 | References | Similar Articles | Additional Information

Abstract: If is a uniformly bounded semigroup on a complex Banach space , then , , generates a holomorphic semigroup on , and is subordinated to 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.

**[1]**A. V. Balakrishnan,*An operational calculus for infinitesimal generators of semigroups.*, Trans. Amer. Math. Soc.**91**(1959), 330–353. MR**0107179**, https://doi.org/10.1090/S0002-9947-1959-0107179-0**[2]**A. V. Balakrishnan,*Fractional powers of closed operators and the semigroups generated by them*, Pacific J. Math.**10**(1960), 419–437. MR**0115096****[3]**S. Bochner,*Diffusion equation and stochastic processes*, Proc. Nat. Acad. Sci. U. S. A.**35**(1949), 368–370. MR**0030151****[4]**Salomon Bochner,*Harmonic analysis and the theory of probability*, University of California Press, Berkeley and Los Angeles, 1955. MR**0072370****[5]**W. Feller,*An introduction to probability theory and its applications*, vol. 2, 2nd ed., Wiley, New York, 1971.**[6]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[7]**Edward Nelson,*A functional calculus using singular Laplace integrals*, Trans. Amer. Math. Soc.**88**(1958), 400–413. MR**0096136**, https://doi.org/10.1090/S0002-9947-1958-0096136-8**[8]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697****[9]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[10]**R. S. Phillips,*On the generation of semigroups of linear operators*, Pacific J. Math.**2**(1952), 343–369. MR**0050797****[11]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528****[12]**Kôsaku Yosida,*An abstract analyticity in time for solutions of a diffusion equation.*, Proc. Japan Acad.**35**(1959), 109–113. MR**0105559****[13]**Kôsaku Yosida,*Fractional powers of infinitesimal generators and the analyticity of the semi-groups generated by them*, Proc. Japan Acad.**36**(1960), 86–89. MR**0121665****[14]**Kôsaku Yosida,*On a class of infinitesimal generators and the integration problem of evolution equations*, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif, 1961, pp. 623–633. MR**0137009****[15]**Kôsaku Yosida,*On holomorphic Markov processes*, Proc. Japan Acad.**42**(1966), 313–317. MR**0207048**

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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