Complex Analysis and Operator Theory

, Volume 11, Issue 3, pp 491–505 | Cite as

On Chaoticity of the Sum of Chaotic Shifts with Their Adjoints in Hilbert Space and Applications to Some Weighted Shifts Acting on Some Fock–Bargmann Spaces

  • Abdelkader Intissar
  • Jean-Karim Intissar


The aim in this paper is to give sufficient conditions for an operator of the form \(\mathbb {T} + \mathbb {T}^{*}\) to be chaotic (in the sense of Devaney). Here \(\mathbb {T}\) is assumed to be a weighted chaotic shift operator on a Hilbert space and \(\mathbb {T}^{*}\) is its adjoint. Sufficient conditions on the weight sequence are given and then applied to some particular bakward shift unbounded operators realized as differential operators acting on some Fock–Bargmann spaces.


Chaotic operators Weighted shift unbounded operators  Analytic functions Fock–Bargmann spaces Gelfond–Leontiev operators Spectral analysis 

Mathematics Subject Classification

46C 47B 47F 32K 


  1. 1.
    Ansari, S.I.: Hypercyclic and cyclic vectors. J. Funct. Anal. 128(2), 374–383 (1995)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Mon. 99, 332–334 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bargmann, V.: On Hilbert space of analytic functions and associated integral transform. Part I. Commun. Pure Appl. Math. 14, 187–214 (1961)CrossRefMATHGoogle Scholar
  4. 4.
    Berezanskii, Y.M.: Expansion in Eigenfunctions of Selfadjoint Operators. American Mathematical Society, Providence, RI (1968)MATHGoogle Scholar
  5. 5.
    Bermudez, T., Bonilla, A., Torrea, J.L.: Chaotic behavior of the Riesz transforms for Hermite expansions. J. Math. Anal. Appl. 337, 702–711 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bès, J., Chan, K., Seubert, S.: Chaotic unbounded differentiation operators. Integr. Eqs. Oper. Theory 40, 257–267 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Birkhoff, G.D.: Démonstration d’un théorème élémentaire sur les fonctions entières. C. R. Acad. Sci. Paris 189, 473–475 (1929)MATHGoogle Scholar
  8. 8.
    Decarreau, A., Emamirad, H., Intissar, A.: Chaoticité del’opératur de Gribov dans l’espace de Bargmann. C. R. Acad. Sci. Paris 331, 751–756 (2000)CrossRefMATHGoogle Scholar
  9. 9.
    Gelfond, A.O., Leontiev, A.F.: On a generalization of the Fourier series. Math. Stab. 29(3), 477–500 (1951)MathSciNetGoogle Scholar
  10. 10.
    Ghanmi, A., Intissar, A.: Construction of concrete orthonormal basis for \((L^{2}, \Gamma, \chi )\)-theta functions associated to discrete subgroups of rank one in \((\mathbb{C},+)\). J. Math. Phys. 54, 063514 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Godefroy, G., Shapiro, M.: Operators with dense, invariant cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Grosse-Erdmann, K.G.: Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36, 345–381 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Grosse-Erdmann, K.G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139, 47–68 (2000)MathSciNetMATHGoogle Scholar
  14. 14.
    Gulisashvili, A., MacCluer, C.: Linear chaos in the unforced quantum harmonic oscillator. J. Dyn. Syst. Meas. Control 118, 337–338 (1996)CrossRefMATHGoogle Scholar
  15. 15.
    Intissar, A.: On a chaotic weighted shift\(z^{p}\frac{d^{p+1}}{dz^{p+1}}\) of order \(p\) in Bargmann space. Adv. Math. Phys. 2011, 471314 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Intissar, A.: A short note on the chaoticity of a weight shift onconcrete orthonormal basis associated to some Fock-Bargmann space. J. Math. Phys 55, 011502 (2014). doi: 10.1063/1.4861931 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Intissar, A.: On the chaoticity of some tensor product weighted backward shiftoperators acting on some tensor product Fock-Bargmann spaces. Complex Anal. Oper. Theory (2015). doi: 10.1007/s11785-015-0459-6 MATHGoogle Scholar
  18. 18.
    Intissar, A.: Analyse de scattering d’un op\(\acute{e}\)rateur cubique de Heun dans l’espace de Bargmann. Commun. Math. Phys. 199, 243–256 (1998)CrossRefGoogle Scholar
  19. 19.
    Intissar, A.: Etude spectrale d’une famille d’opérateurs non symétriques intervenant dans la théorie des champs des reggeons. Commun. Math. Phys. 113(2), 263–297 (1987)CrossRefMATHGoogle Scholar
  20. 20.
    Intissar, A.: Spectral analysis of non-self-adjoint Jacobi–Gribov operator and asymptotic analysis of its generalized eigenvectors. Adv. Math. (China) 44(3), 335–353 (2015). doi: 10.11845/sxjz.2013117b MATHGoogle Scholar
  21. 21.
    Intissar, A.: On a chaotic weighted shift \(z^{p}\mathbb{D}^{p+1}\) of order \(p\) in generalized Fock-Bargmann spaces. Math. Aeterna 3(7), 519–534 (2013)MathSciNetMATHGoogle Scholar
  22. 22.
    Intissar, A.: On chaoticity of sum of weithed shift \(z^{p}{{D}}^{p+1}\) of order \(p\) with its adjoint in generalized Bargmann space. (submitted) (2016)Google Scholar
  23. 23.
    Irac-Astaud, M., Rideau, G.: From Bargmann representations to deformed harmonic oscillator algebras. Rep. Math. Phys. 43(112), 1 (1999)MathSciNetMATHGoogle Scholar
  24. 24.
    Maclane, G.R.: Sequences of derivatives and normal families. J. Anal. Math. 2, 72–87 (1952)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Milne-Thomson, L.M.: The Calculus of Finite Differences. MacMillan, London (1951)MATHGoogle Scholar
  26. 26.
    Montel, P.: Leçons sur les Recurrences et Leur Applications. Gauthier-Villars, Paris (1957)MATHGoogle Scholar
  27. 27.
    Salas, H.N.: Pathological hypercyclic operators. Arch. Math. 86, 241–250 (2006)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Equipe d’Analyse spectrale, Faculté des Sciences et TechniquesUniversité de CortéCortéFrance
  2. 2.Ecole Saint-GenevièveAcadémie de VersaillesVersaillesFrance
  3. 3.Le PradorMarseilleFrance

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