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1998 | 129 | 2 | 137-156
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On analytic semigroups and cosine functions in Banach spaces

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If A generates a bounded cosine function on a Banach space X then the negative square root B of A generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by B. The characterization relies on new results on the inversion of the vector-valued conjugate potential transform.
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  • Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger Strasse, 67663 Kaiserslautern, Germany, vieten@mathematik.uni-kl.de
Bibliografia
  • [1] W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
  • [2] W. Arendt, personal communication.
  • [3] W. Arendt and H. Kellermann, Integrated solutions of Volterra integrodifferential equations and applications, in: Volterra Integrodifferential Equations in Banach Spaces and Applications (Proc. Conf. Trento 1987), G. Da Prato and M. Iannelli (eds.), Pitman Res. Notes Math. Ser. 190, Longman Sci. Tech., Harlow, 1989, 21-51.
  • [4] J. W. Dettman, Initial-boundary value problems related through the Stieltjes transform, J. Math. Anal. Appl. 25 (1969), 341-349.
  • [5] O. El Mennaoui and V. Keyantuo, Trace theorems for holomorphic semigroups and the second order Cauchy problem, Proc. Amer. Math. Soc. 124 (1996), 1445-1458.
  • [6] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland, Amsterdam, 1985.
  • [7] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monographs, Oxford Univ. Press, New York, 1985.
  • [8] E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, 1975.
  • [9] M. Hieber, Integrated semigroups and differential operators on $L^p(ℝ^N)$-spaces, Math. Ann. 291 (1991), 1-16.
  • [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc. Providence, R.I., 1957.
  • [11] S. Kurepa, A cosine functional equation in Banach algebras, Acta Sci. Math. (Szeged) 23 (1962), 255-267.
  • [12] H. R. Thieme, Integrated semigroups and integrated solutions to the abstract Cauchy problem, J. Math. Anal. Appl. 152 (1990), 416-447.
  • [13] P. Vieten, Holomorphie und Laplace Transformation Banachraumwertiger Funktionen, Ph.D. thesis, Shaker, Aachen, 1995.
  • [14] D. V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971.
  • [15] K. Yosida, Functional Analysis, Springer, New York, 1980.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv129i2p137bwm
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