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Subordination in the sense of Bochner and a related functional calculus

Part of: General theory of linear operators Markov processes Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  09 April 2009

René L. Schilling
René L. Schilling
Affiliation:
Max-Planck-InstitutFür Mathematik in den Naturwissenschaften Inselstrasse 22-26 04103 LeipzigGermany
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Abstract

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We prove a new representation of the generator of a subordinate semigroup as limit of bounded operators. Our construction yields, in particular, a characterization of the domain of the generator. The generator of a subordinate semigroup can be viewed as a function of the generator of the original semigroup. For a large class these functions we show that operations at the level of functions has its counterpart at the level of operators.

Keywords

strongly continuous semigroupsubordinationfunctional calculusfunctions of an operator.

MSC classification

Secondary: 47D03: Groups and semigroups of linear operators 47A60: Functional calculus 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 368 - 396
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Berg, C., ‘The Stieltjes cone is logarithmically convex’, in: Laine, I., Lehto, O. and Sorvali, T., Complex analysis, Joensuu 1978, Lecture Notes in Math. 747, (Springer, Berlin, 1979), pp. 46–54.Google Scholar
[2]Berg, C., Boyadzhiev, Kh. and deLaubenfels, R., ‘Generation of generators of holomorphic semigroups’, J. Austral. Math. Soc. (Series A) 55 (1993), 246–269.CrossRefGoogle Scholar
[3]Berg, C. and Forst, G., Potential theory on locally compact Abelian groups, Ergeb. Math. Grenzgeb. II, 87 (Springer, Berlin, 1975).Google Scholar
[4]Bochner, S., ‘Diffusion equation and stochastic processes’, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 368–370.Google Scholar
[5]Bochner, S., Harmonic analysis and the theory of probability (University of Califormia Press, Berkeley, 1955).CrossRefGoogle Scholar
[6]deLaubenfels, R., ‘Automatic extensions of functional calculi’, Studia Math. 114 (1995), 237–259.Google Scholar
[7]Dunford, N. and Schwartz, J. T., Linear operators I, Pure Appl. Math. 7 (Interscience, New York, 1957).Google Scholar
[8]Ethier, S. E. and Kurtz, T. G., Markov processes: Characterization and convergence, Wiley Ser. Probab. Math. Statist. (Wiley, New York 1986).Google Scholar
[9]Heinz, E., ‘Beiträge zur Störungstheorie de Spektralzerlegung’, Math. Ann. 123 (1951), 415–438.CrossRefGoogle Scholar
[10]Hille, E. and Phillips, R. S., Functional analyis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 (Amer. Math. Soc., Providence, 1957) (2nd ed.).Google Scholar
[11]Hirsch, F., ‘Domaines d'opérateurs representés comme integrales de resolvantes’, J. Funct. Anal. 23 (1976), 239–264.CrossRefGoogle Scholar
[12]Jacob, N. and Schilling, R. L., ‘Subordination in the sense of S. Bochner - An approach through pseudo differential operators’, Math. Nachr. 178 (1996), 199–231.Google Scholar
[13]Nollau, V., ‘Über gebrochene Potenzen infinitesimaler Generatoren Markovscher Übergangswahrscheinlichkeiten I’, Math. Nachr. 65 (1975), 235–246.Google Scholar
[14]Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied Math. Ser. 44 (Springer, Berlin, 1983).Google Scholar
[15]Phillips, R. S., ‘On the generation of semigroups of linear operators’, Pacific J. Math. 2 (1952), 343–369.Google Scholar
[16]Prüss, J., Evolutionary integral equations and applications, Monographs Math. 87 (Birkhaüser, Basel, 1993).CrossRefGoogle Scholar
[17]Rogers, L. C. G. and Williams, D., Diffusions, Markov processes, and martingales. I: Foundations, Wiley Ser. Probab. Math. Statist. (Wiley, New York, 1994) (2nd ed.).Google Scholar
[18]Schilling, R. L., Zum Pfadverhalten von Markovschen Prozessen, die mit Lévy-Prozessen vergleichbar sind (Dissertation, Universität Erlangen 1994).Google Scholar
[19]Schilling, R. L., ‘On the domain of the generator of a subordinate semigroup’, in: Král, J. et al. , (eds.), Potential Theory—ICPT 94. Proceedings Internat. Conf Potential Theory, Kouty (CR), 1994 (de Gruyter, Berlin, 1996), pp. 449–462.Google Scholar
[20]Srivastava, H. M. and Tuan, V. K., ‘A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations’, Arch. Math. 64 (1995), 144–149.CrossRefGoogle Scholar
[21]Tanabe, H., Equations of evolution, Pitman Monographs Surveys Pure Appl. Math. 6 (Longman Sci. Tech., Harlow, 1979).Google Scholar
[22]Westphal, U., ‘Ein kalküll für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren. Teil I: Halbgruppenerzeuger’, Compositio Math. 22 (1970), 67–103.Google Scholar
[23]yosida, K., Functional analysis, Grundlehren Math. Wiss. 123 (Springer, Berlin, 1980) (6th ed.).Google Scholar