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On generators of C0-semigroups of composition operators

  • Eva A. Gallardo-Gutiérrez
  • Dmitry V. Yakubovich
Article
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Abstract

Avicou, Chalendar and Partington proved in 2015 [5] that an (unbounded) operator Af = G·f' on the classical Hardy space generates a C0 semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator A generates a C0 semigroup, then it is automatically a semigroup of composition operators, so that the condition of quasicontractivity of the semigroup in the cited result is not necessary. Our result applies to a rather general class of Banach spaces of analytic functions in the unit disc.

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References

  1. [1]
    A. Anderson, M. Jovovic and W. Smith, Composition semigroups on BMOA and H∞, Journal of Mathematical Analysis and Applications 449 (2017), 843–852.MathSciNetCrossRefGoogle Scholar
  2. [2]
    W. Arendt, Ch. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, Vol. 96, Birkhäuser/Springer, Basel, 2011.Google Scholar
  3. [3]
    W. Arendt and I. Chalendar, Generators of semigroups on Banach spaces inducing holomorphic semiflows, Israel Journal of Mathematics, to appear, arXiv:1803.06552v1.Google Scholar
  4. [4]
    I. Arévalo, M. D. Contreras and L. Rodríguez-Piazza, Semigroups of composition operators and integral operators on mixed norm spaces, arXiv:1610.08784.Google Scholar
  5. [5]
    C. Avicou, I. Chalendar and J. R. Partington, JA class of quasicontractive semigroups acting on Hardy and Dirichlet space, ournal of Evolution Equations 15 (2015), 647–665.CrossRefGoogle Scholar
  6. [6]
    C. Avicou, I. Chalendar and J. R. Partington, Analyticity and compactness of semigroups of composition operators, Journal of Mathematical Analysis and Applications 437 (2016), 545–560.MathSciNetCrossRefGoogle Scholar
  7. [7]
    C. Berenstein and R. Gay, Complex Analysis and Topics in Harmonic Analysis, Springer, New York, 1995.CrossRefGoogle Scholar
  8. [8]
    E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Mathematical Journal 25 (1978), 101–115.MathSciNetCrossRefGoogle Scholar
  9. [9]
    O. Blasco, M. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis and A. Siskakis, Semigroups of composition operators and integral operators in spaces of analytic functions, Annales Academiæ Scientiarum Fennicæ. Mathematica 38 (2013), 67–89.MathSciNetCrossRefGoogle Scholar
  10. [10]
    P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Transactions of the American Mathematical Society 351 (1999), 2183–2196.MathSciNetCrossRefGoogle Scholar
  11. [11]
    I. Chalendar and J. R. Partington, Norm estimates for weighted composition operators on spaces of holomorphic functions, Complex Analysis and Operator Theory 8 (2014), 1087–1095.MathSciNetCrossRefGoogle Scholar
  12. [12]
    I. Chalendar and J. R. Partington, A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces, Semigroup Forum 95 (2017), 281–292.MathSciNetCrossRefGoogle Scholar
  13. [13]
    C. Cowen, An application of Hadamard multiplication to operators on weighted Hardy spaces, Linear Algebra and its Applications 133 (1990), 21–32.MathSciNetCrossRefGoogle Scholar
  14. [14]
    C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton,FL, 1995.zbMATHGoogle Scholar
  15. [15]
    O. El-Fallah, K. Kellay, J. Mashreghi and T. Ransford, A Primer on the Dirichlet Space, Cambridge Tracts in Mathematics, Vol. 203, Cambridge University Press, Cambridge, 2014.zbMATHGoogle Scholar
  16. [16]
    K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Text in Mathematics, Vol. 194, Springer, New York, 2000.Google Scholar
  17. [17]
    E. A. Gallardo-Gutiérrez and A. Montes-Rodríguez, Adjoints of linear fractional composition operators on the Dirichlet space, Mathematische Annalen 327 (2003), 117–134.MathSciNetCrossRefGoogle Scholar
  18. [18]
    E. A. Gallardo-Gutiérrez and J. R. Partington, Norms of composition operators on weighted Hardy spaces, Israel Journal of Mathematics 196 (2013), 273–283.MathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, Vol. 169, Birkhäuser, Basel, 2006.Google Scholar
  20. [20]
    E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley-Interscience, New York–London–Sidney, 1976.zbMATHGoogle Scholar
  21. [21]
    V. É. Kacnel’son, A remark on canonical factorization in certain spaces of analytic functions. Investigations on linear operators and the theory of functions, III, Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR 30 (1972), 163–164; English translation: Journal of Soviet Mathematics 4 (1975), 444–445.Google Scholar
  22. [22]
    M. J. Martín and D. Vukotic, Norms and spectral radii of composition operators acting on the Dirichlet space, Journal of Mathematical Analysis and its Applications 304 (2005), 22–32.MathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Matolcsi, On the relation of closed forms and Trotter’s product formula, Journal of Functional Analysis 205 (2003), 401–413.MathSciNetCrossRefGoogle Scholar
  24. [24]
    M. Matolcsi, On quasi-contractivity of C0-semigroups on Banach spaces, Archiv der Mathematik 83 (2004), 360–363.MathSciNetCrossRefGoogle Scholar
  25. [25]
    S. Reich and D. Shoikhet, Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.zbMATHGoogle Scholar
  26. [26]
    D. Shoikhet, Semigroups in Geometrical Function Theory, Kluwer, Dordrecht, 2001.CrossRefGoogle Scholar
  27. [27]
    A. G. Siskakis, Semigroups of composition operators on the Dirichlet space, Results in Mathematics 30 (1996), 165–173.MathSciNetCrossRefGoogle Scholar
  28. [28]
    A. G. Siskakis, Semigroups of composition operators on spaces of analytic functions, areview, in Studies on Composition Operators (Laramie, WY, 1996), Contemporary Mathematics, Vol. 213, American Mathematical Society, Providence, RI, 1998, pp. 229–252.zbMATHGoogle Scholar

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Eva A. Gallardo-Gutiérrez
    • 1
    • 2
  • Dmitry V. Yakubovich
    • 3
    • 4
  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain
  3. 3.Departamento de MatemáticasUniversidad Autónoma de Madrid, CantoblancoMadridSpain
  4. 4.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

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