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.
Similar content being viewed by others
References
A. Aleman and J. Cima,An integral operator on H p and Hardy's inequality, J. Analyse Math.85 (2001), 157–176.
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001.
S. Axler, P. Bourdon and W. Ramey,Harmonic Function Theory, Springer-Verlag, New York, 2001.
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.
C. J. K. Batty, R. Chill and Y. Tomilov,Strong stability of bounded evolution families and semigroups, J. Funct. Anal.193 (2002), 116–139.
A. Beurling,Analytic continuation across a linear boundary, Acta Math.128 (1972), 153–182.
A. Borichev and H. Hedenmalm,Completeness of translates in weighted spaces on the half-line, Acta Math.174 (1995), 1–84.
K. N. Boyadzhiev and N. Levan,Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung.30 (1995), 165–182.
H. Bremermann,Distributions, Complex Variables, and Fourier Transforms, Addison-Wesley, Reading, Mass., 1965.
T. Carleman,L'intégrale de Fourier et questions que s'y rattachent, Publications Scientifiques de l'Institut Mittag-Leffler, Vol. 1, Uppsala, 1944.
F. W. Carroll and D. J. Troy,Distributions and analytic continuation, J. Analyse Math.24 (1971), 87–100.
R. Chill and Y. Tomilov,Stability of C 0-semigroups and geometry of Banach spaces, Math. Proc. Cambridge Philos. Soc.135 (2003), 493–511.
B. E. J. Dahlberg,On the radial boundary values of subharmonic functions, Math. Scand.40 (1977), 301–317.
B. E. J. Dahlberg,On the Poisson integral for Lipschitz and C 1-domains, Studia Math.66 (1979), 13–24.
L. Ehrenpreis,Reflection, removable singularities, and approximation for partial differential equations. 1, Ann. of Math.112 (1980), 1–20.
L. Ehrenpreis,Reflection, removable singularities, and approximation for partial differential equations. II, Trans. Amer. Math. Soc.302 (1987), 1–45.
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.
P. C. Fenton,Line integrals of subharmonic functions, J. Math. Anal. Appl.168 (1992), 108–110.
R. M. Gabriel,Some inequalities concerning integrals of two-dimensional and three-dimensional subharmonic functions, J. London Math. Soc.24 (1949), 313–316.
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.
F. Gehring,Variations on a theorem of Fejér and Riesz, Ann. Univ. Mariae Curie-Sklodowska Sect. A53 (1999), 57–66.
A. Granados,On a problem raised by Gabriel and Beurling, Michigan Math. J.46 (1999), 461–487.
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.
R. Harvey and J. Polking,Removable singularities of solutions of linear partial differential equations, Acta Math.125 (1970), 39–56.
V. Havin and B. Jöricke,The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994.
L. Hörmander,L p estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967), 65–78.
S. V. Hruščëv,Sets of uniqueness for the Gevrey classes, Ark. Mat.15 (1977), 253–304.
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.
Y. Katznelson,An Introduction to Harmonic Analysis, 2nd corrected edition, Dover Publications, New York, 1976.
K. Kelley,Contractions et hyperdistributions à spectre de Carleson, J. London Math. Soc.58 (1998), 185–196.
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.
H. Komatsu,Fractional powers of operators IV: potential operators, J. Math. Soc. Japan21 (1969), 221–228.
C. Martinez and M. Sanz,The Theory of Fractional Powers of Operators, North-Holland, Amsterdam, 2001.
J. M. A. M. van Neerven,The Asymptotic Behaviour of Semigroups of Linear Operators, Birkhäuser Verlag, Basel, 1996.
N. K. Nikolski,Treatise on the Shift Operator. Spectral Function Theory, Springer-Verlag, Berlin, 1986.
A. V. Noell and T. Wolff,Peak sets for Lip α classes, J. Funct. Anal.86 (1989), 136–179.
J. Prüss,Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.
W. Rudin,Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, R.I., 1971.
B. Sz.-Nagy and C. Foiaş,Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.
Y. Tomilov,A resolvent approach to stability of operator semigroups, J. Operator Theory46 (2001), 63–98.
Quôc Phong Vũ,On the spectrum, complete trajectories and asymptotic stability of linear semidynamical systems, J. Differential Equations105 (1993), 30–45.
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.
F. Wolf,The Poisson integral. A study in the uniqueness of functions, Acta Math.74 (1941), 65–100.
F. Wolf,Extension of analytic functions, Duke Math. J.14 (1947), 877–887.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
Chill, R., Tomilov, Y. Analytic continuation and stability of operator semigroups. J. Anal. Math. 93, 331–357 (2004). https://doi.org/10.1007/BF02789312
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02789312