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Analytic continuation and stability of operator semigroups

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Abstract

We prove a new criterion for the analytic continuation of functions across a linear boundary. As corollaries, we obtain new conditions for convergence of orbits of operator semigroups on Banach spaces with Fourier type.

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References

  1. A. Aleman and J. Cima,An integral operator on H p and Hardy's inequality, J. Analyse Math.85 (2001), 157–176.

    MATH  MathSciNet  Google Scholar 

  2. W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001.

    MATH  Google Scholar 

  3. S. Axler, P. Bourdon and W. Ramey,Harmonic Function Theory, Springer-Verlag, New York, 2001.

    MATH  Google Scholar 

  4. C. J. K. Batty, Z. Brzezniak and D. A. Greenfield,A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum, Studia Math.121 (1996), 167–182.

    MATH  MathSciNet  Google Scholar 

  5. C. J. K. Batty, R. Chill and Y. Tomilov,Strong stability of bounded evolution families and semigroups, J. Funct. Anal.193 (2002), 116–139.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Beurling,Analytic continuation across a linear boundary, Acta Math.128 (1972), 153–182.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Borichev and H. Hedenmalm,Completeness of translates in weighted spaces on the half-line, Acta Math.174 (1995), 1–84.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. N. Boyadzhiev and N. Levan,Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung.30 (1995), 165–182.

    MATH  MathSciNet  Google Scholar 

  9. H. Bremermann,Distributions, Complex Variables, and Fourier Transforms, Addison-Wesley, Reading, Mass., 1965.

    MATH  Google Scholar 

  10. T. Carleman,L'intégrale de Fourier et questions que s'y rattachent, Publications Scientifiques de l'Institut Mittag-Leffler, Vol. 1, Uppsala, 1944.

  11. F. W. Carroll and D. J. Troy,Distributions and analytic continuation, J. Analyse Math.24 (1971), 87–100.

    MATH  MathSciNet  Google Scholar 

  12. R. Chill and Y. Tomilov,Stability of C 0-semigroups and geometry of Banach spaces, Math. Proc. Cambridge Philos. Soc.135 (2003), 493–511.

    Article  MATH  MathSciNet  Google Scholar 

  13. B. E. J. Dahlberg,On the radial boundary values of subharmonic functions, Math. Scand.40 (1977), 301–317.

    MATH  MathSciNet  Google Scholar 

  14. B. E. J. Dahlberg,On the Poisson integral for Lipschitz and C 1-domains, Studia Math.66 (1979), 13–24.

    MATH  MathSciNet  Google Scholar 

  15. L. Ehrenpreis,Reflection, removable singularities, and approximation for partial differential equations. 1, Ann. of Math.112 (1980), 1–20.

    Article  MathSciNet  Google Scholar 

  16. L. Ehrenpreis,Reflection, removable singularities, and approximation for partial differential equations. II, Trans. Amer. Math. Soc.302 (1987), 1–45.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Esterle, M. Zarrabi and M. Rajoelina,On contractions with spectrum contained in the Cantor set, Math. Proc. Cambridge Philos. Soc.117 (1995), 339–343.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. C. Fenton,Line integrals of subharmonic functions, J. Math. Anal. Appl.168 (1992), 108–110.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. M. Gabriel,Some inequalities concerning integrals of two-dimensional and three-dimensional subharmonic functions, J. London Math. Soc.24 (1949), 313–316.

    Article  MathSciNet  Google Scholar 

  20. J. Garcia-Cuerva, K. S. Kazaryan, V. I. Kolyada and Kh. L. Torrea,The Hausdorff-Young inequality with vector-valued coefficients and applications, Uspekhi Mat. Nauk53 (1998), 3–84; Engl. transl.: Russian Math. Surveys53 (1998), 435–513.

    MathSciNet  Google Scholar 

  21. F. Gehring,Variations on a theorem of Fejér and Riesz, Ann. Univ. Mariae Curie-Sklodowska Sect. A53 (1999), 57–66.

    MATH  MathSciNet  Google Scholar 

  22. A. Granados,On a problem raised by Gabriel and Beurling, Michigan Math. J.46 (1999), 461–487.

    Article  MATH  MathSciNet  Google Scholar 

  23. V. P. Gurarii,Harmonic analysis in spaces with a weight, Trudy Moskov. Mat. Obsch.35 (1976), 21–76; Engl. Transl.: Trans. Moscow Math. Soc.1 (1979), 21–75.

    MathSciNet  Google Scholar 

  24. R. Harvey and J. Polking,Removable singularities of solutions of linear partial differential equations, Acta Math.125 (1970), 39–56.

    MATH  MathSciNet  Google Scholar 

  25. V. Havin and B. Jöricke,The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994.

    MATH  Google Scholar 

  26. L. Hörmander,L p estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967), 65–78.

    MATH  MathSciNet  Google Scholar 

  27. S. V. Hruščëv,Sets of uniqueness for the Gevrey classes, Ark. Mat.15 (1977), 253–304.

    Article  MathSciNet  Google Scholar 

  28. S. V. Hruščëv,The problem of simultaneous approximation and of removal of the singularities of Cauchy type integrals, inSpectral Theory of Functions and Operators, Proc. Steklov Inst. Math.4 (1979), Amer. Math. Soc. Transl., Providence, R.I., pp. 133–203.

    Google Scholar 

  29. Y. Katznelson,An Introduction to Harmonic Analysis, 2nd corrected edition, Dover Publications, New York, 1976.

    MATH  Google Scholar 

  30. K. Kelley,Contractions et hyperdistributions à spectre de Carleson, J. London Math. Soc.58 (1998), 185–196.

    Article  MathSciNet  Google Scholar 

  31. C. Kenig,Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conf. Ser. in Math., Vol. 83, Amer. Math. Soc., Providence, R.I., 1994.

    MATH  Google Scholar 

  32. H. Komatsu,Fractional powers of operators IV: potential operators, J. Math. Soc. Japan21 (1969), 221–228.

    Article  MathSciNet  Google Scholar 

  33. C. Martinez and M. Sanz,The Theory of Fractional Powers of Operators, North-Holland, Amsterdam, 2001.

    MATH  Google Scholar 

  34. J. M. A. M. van Neerven,The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser Verlag, Basel, 1996.

    MATH  Google Scholar 

  35. N. K. Nikolski,Treatise on the Shift Operator. Spectral Function Theory, Springer-Verlag, Berlin, 1986.

    Google Scholar 

  36. A. V. Noell and T. Wolff,Peak sets for Lip α classes, J. Funct. Anal.86 (1989), 136–179.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Prüss,Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.

    MATH  Google Scholar 

  38. W. Rudin,Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, R.I., 1971.

    MATH  Google Scholar 

  39. B. Sz.-Nagy and C. Foiaş,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.

    Google Scholar 

  40. Y. Tomilov,A resolvent approach to stability of operator semigroups, J. Operator Theory46 (2001), 63–98.

    MathSciNet  Google Scholar 

  41. Quôc Phong Vũ,On the spectrum, complete trajectories and asymptotic stability of linear semidynamical systems, J. Differential Equations105 (1993), 30–45.

    Article  MathSciNet  MATH  Google Scholar 

  42. U. Westphal,A generalized version of the abelian mean ergodic theorem with rates for semigroup operators and fractional powers of infinitesimal generators, Results Math.34 (1998), 381–394.

    MATH  MathSciNet  Google Scholar 

  43. F. Wolf,The Poisson integral. A study in the uniqueness of functions, Acta Math.74 (1941), 65–100.

    Article  MATH  MathSciNet  Google Scholar 

  44. F. Wolf,Extension of analytic functions, Duke Math. J.14 (1947), 877–887.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Ralph Chill.

Additional information

This work started during a visit of the first author at the University of Torun. The kind hospitality is gratefully acknowledged. The second author was partially supported by a KBN grant and by the NASA-NSF Twinning Program.

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Chill, R., Tomilov, Y. Analytic continuation and stability of operator semigroups. J. Anal. Math. 93, 331–357 (2004). https://doi.org/10.1007/BF02789312

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  • DOI: https://doi.org/10.1007/BF02789312

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