Israel Journal of Mathematics

, Volume 229, Issue 1, pp 487–500 | Cite as

On generators of C0-semigroups of composition operators

  • Eva A. Gallardo-GutiérrezEmail author
  • Dmitry V. Yakubovich


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. Berenstein and R. Gay, Complex Analysis and Topics in Harmonic Analysis, Springer, New York, 1995.CrossRefzbMATHGoogle Scholar
  8. [8]
    E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Mathematical Journal 25 (1978), 101–115.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M. Matolcsi, On the relation of closed forms and Trotter’s product formula, Journal of Functional Analysis 205 (2003), 401–413.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Matolcsi, On quasi-contractivity of C0-semigroups on Banach spaces, Archiv der Mathematik 83 (2004), 360–363.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle Scholar
  27. [27]
    A. G. Siskakis, Semigroups of composition operators on the Dirichlet space, Results in Mathematics 30 (1996), 165–173.MathSciNetCrossRefzbMATHGoogle 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

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Eva A. Gallardo-Gutiérrez
    • 1
    • 2
    Email author
  • Dmitry V. Yakubovich
    • 3
    • 2
  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

Personalised recommendations