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Generation of generators of holomorphic semigroups

Part of: Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Christian Berg ,
Khristo Boyadzhiev  and
Ralph Delaubenfels
Christian Berg
Affiliation:
Matematisk Institut Universitetsparken, 5, DK-2100, Kobenhavnø, Demark
Khristo Boyadzhiev
Affiliation:
Ohio Northern University, Ada, Ohio 45810, USA
Ralph Delaubenfels
Affiliation:
Ohio University, Athens, Ohio 45701, USA
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Abstract

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We construct a functional calculus, g → g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (−∞, 0), with {‖r(r + A)−1‖ ¦ r > 0} bounded. For such functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.

We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever − A does.

We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.

MSC classification

Secondary: 47A60: Functional calculus 47B44: Accretive operators, dissipative operators, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 246 - 269
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Akhiezer, N., The classical moment problem (Hafner Publishing Company, New York, 1965).Google Scholar
[2]Ando, T., ‘Comparison of norms f(A) − and f(¦A - B¦) ’, Math. Z. 197 (1988), 403–409.Google Scholar
[3]Ash, R. B., Measure, integration and functional analysis (Academic Press, New York, 1972).Google Scholar
[4]Balakrishnan, A. V., ‘Fractional powers of closed operators and the semigroups generated by them’, Pac. J. Math. 10 (1960), 419–437.Google Scholar
[5]Berg, C., Quelques remarques sur le cone de Stieltjes, Lecture Notes in Math. 814 (Springer, Berlin, 1980) pp. 70–79.Google Scholar
[6]Berg, C., Christensen, J. P. R. and Ressel, P., Harmonic Analysis on Semigroups (Springer, Berlin, 1984).Google Scholar
[7]Berg, C. and Forst, G., Potential theory on locally compact abelian groups (Springer, Berlin, 1975).CrossRefGoogle Scholar
[8]Bochner, S., ‘Diffusion equations and stochastic processes’, Proc. Nat. Acad. Soc. U.S.A 35 (1949), 368–370.Google Scholar
[9]Bochner, S., Harmonic analysis and the theory of probability (University of California Press, Berkeley, 1955).Google Scholar
[10]Boyadzhiev, K. and deLaubenfels, R., ‘h∞-functional calculus for perturbations of generators of holomorphic semigroups’, Houston J. Math. 17 (1991), 131–147.Google Scholar
[11]Bratteli, O., Kishimoto, A. and Robinson, D. W., ‘Positivity and monotonicity properties of C0 semigroups I’, Comm. Math. Phys. 75 (1980), 67–84.Google Scholar
[12]Carasso, A. S. and Kato, T., ‘On subordinated holomoprhic semigroups’, Trans. Amer. Math. Soc. 327 (1991), 867–878.Google Scholar
[13]Davies, E. B., One-parameter semigroups (Academic Press, London, 1980).Google Scholar
[14]deLaubenfels, R., ‘C-semigroups and the Cauchy problem’, J. Funct. Anal., to appear.Google Scholar
[15]deLaubenfels, R., ‘Functional calculus and abstract Cauchy problems for operators with polynomially bounded resolvent’, J. Funct. Anal., to appear.Google Scholar
[16]deLaubenfels, R., ‘Powers of generators of holomorphic semigroups’, Proc. Amer. Math. Soc. 99 (1987), 105–108.Google Scholar
[17]deLaubenfels, R., ‘Inverses of generators’, Proc. Amer. Math. Soc. 104 (1988), 443–448.Google Scholar
[18]Donoghue, W. F. Jr, Monotone matrix functions and analytic continuation (Springer-Verlag, New York, 1974).Google Scholar
[19]Dunford, N. and Schwartz, J. T., Linear operators, part I (Interscience, New York, 1958).Google Scholar
[20]Duong, X. T., ‘H∞ functional calculus of elliptic operators with C∞ coefficients on Lp spaces of smooth domains’, J. Austral. Math. Soc. (Series A) 48 (1990), 113–123.Google Scholar
[21]Fattorini, H. O., The abstract Cauchy problem (Addison Wesley, Reading, 1983).Google Scholar
[22]Goldstein, J. A., Semigroups of operators and applications (Oxford, New York, 1985).Google Scholar
[23]Heinz, E., ‘Beitrage zur Störungstheorie der Spektralzerlegung’, Math. Ann. 123 (1951), 415–438.Google Scholar
[24]Hille, E. and Phillips, R. S., Functional analysis and semigroups (Amer. Math. Soc., Providence, 1957).Google Scholar
[25]Hirsch, F., ‘Integrales de resolvantes et calcul symbolique’, Ann. Inst. Fourier, Grenoble 22 (4) (1972), 239–264.Google Scholar
[26]Hirsch, F., ‘Domaines d'operateurs representes comme integrales de resolvantes’, J. Funct. Anal. 23 (1976), 199–217.Google Scholar
[27]Kato, T., ‘Note on fractional powers of linear operators’, Proc. Japan Acad. Ser. A Math. Sci. 36 (1960), 94–96.Google Scholar
[28]Kishimoto, A. and Robinson, D. W., ‘Subordinate semigroups and order properties’, J. Austral. Math. Soc.(Series A) 31 (1981), 59–76.CrossRefGoogle Scholar
[29]Komatsu, H., ‘Fractional powers of operators’, Pac. J. Math. 19 (1966), 285–346.Google Scholar
[30]Krein, S. G., Linear differential equations in Banach spaces, Transl. Math. Monogrpahs 29 (Amer. Math. Soc., Providence, 1971).Google Scholar
[31]Lanford, O. E. and Robinson, D. W., ‘Fractional powers of generators of equicontinuous semigroups and fractional derivatives’, J. Austral. Math. Soc.(Series A) 46 (1989), 473–504.Google Scholar
[32]Martinez, C. and Sanz, M., ‘nth roots of a non-negative operators: conditions for uniqueness’, Manuscripta Math. 64 (1989), 403–417.Google Scholar
[33]Martinez, C. and Sanz, M. and Marco, L., ‘Fractional powers of operators’, J. Math. Soc. Japan 40 (1988), 331–347.Google Scholar
[34]McIntosh, A., ‘Operators which have an H∞ functional calculus’, in: Miniconference on Operator Theory and PDE, Proc. of the Center for Math. Analysis (Australian National University, Canberra, 1986) pp. 210–231.Google Scholar
[35]Nakamura, Y., ‘Classes of operator monotone functions and stieltjes functions’, in: Operator Theory: Advances and Applications 41 (Birkhasuer, Basel, 1989) pp. 395–404.Google Scholar
[36]Nelson, E., ‘A functional calculus using singular laplace integrals’, Trans. Amer. Math. Soc. 88 (1959), 400–413.Google Scholar
[37]Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer, New York, 1983).Google Scholar
[38]Phillips, R. S., ‘On the generation of semigroups of linear operators’, Proc. J. Math. 2 (1952), 343–369.Google Scholar
[39]Pustylnik, E. I., ‘On functions of a positive operator’, Math. USSR Sbornik 47 (1984), 27–42.Google Scholar
[40]Reed, M. and Simon, B., Methods of modern mathematical physics II (Academic Press, New York, 1975).Google Scholar
[41]Ricker, W., ‘Spectral properties of the Laplace operator in Lp (R)’, Osaka J. Math. 25 (1988), 399–410.Google Scholar
[42]Ricker, W., ‘An L1-type functional calculus for the Laplace operator in Lp(R)’, J. Operator Theory 21 (1989), 41–67.Google Scholar
[43]Rosenblum, M. and Rovnyak, J., Hardy classes and operator theory (Oxford University Press, Oxford, 1985).Google Scholar
[44]Tanabe, H., Equations of evolution (Pitman, London, 1979).Google Scholar
[45]Watanabe, J., ‘On some properties of fractional powers of linear operators’, Proc. Japan Acad. 37 (1961), 273–275.Google Scholar
[46]Widder, D. V., The Laplace transform (Princeton University Press, Princeton, 1946).Google Scholar
[47]Widder, D. V., Introduction to transform theory (Academic Press, New York and London, 1971).Google Scholar
[48]Yagi, A., ‘Coincidence entre des espaces d'interpolation et des domaines de puissances fractionnaires d'operateurs’, C. R. Acad. Sci. Paris Ser. I Math. 299 (1984), 173–176.Google Scholar
[49]Yosida, K., ‘Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them’, Proc. Japan Acad. Ser. A Math. Sci. 36 (1960), 86–89.Google Scholar
[50]Yosida, K., Functional Analysis (Springer, Berlin, 1978).CrossRefGoogle Scholar