Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 257–274 | Cite as

Dynamics of Generalised Derivations and Elementary Operators

  • Clifford GilmoreEmail author


We identify concrete examples of hypercyclic generalised derivations acting on separable ideals of operators and establish some necessary conditions for their hypercyclicity. We also consider the dynamics of elementary operators acting on particular Banach algebras, which reveals surprising hypercyclic behaviour on the space of bounded linear operators on the Banach space constructed by Argyros and Haydon.


Hypercyclic Generalised derivation Elementary operator 

Mathematics Subject Classification

Primary 47A16 Secondary 47B47 



This article is part of the Ph.D. thesis of the author and he would like to thank his supervisor Hans-Olav Tylli for helpful comments and remarks during its preparation.


  1. 1.
    Anderson, J.: On normal derivations. Proc. Am. Math. Soc. 38, 135–140 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, J., Foiaş, C.: Properties which normal operators share with normal derivations and related operators. Pac. J. Math. 61(2), 313–325 (1975)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Argyros, S.A., Haydon, R.G.: A hereditarily indecomposable \(\cal{L_\infty }\)-space that solves the scalar-plus-compact problem. Acta Math. 206(1), 1–54 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  5. 5.
    Bhatia, R., Rosenthal, P.: How and why to solve the operator equation \(AX-XB=Y\). Bull. Lond. Math. Soc. 29(1), 1–21 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bonet, J., Martínez-Giménez, F., Peris, A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl. 297(2), 599–611 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic subspaces. Monatsh. Math. 168(3–4), 305–320 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bourdon, P.S.: Orbits of hyponormal operators. Mich. Math. J. 44(2), 345–353 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chan, K.C.: Hypercyclicity of the operator algebra for a separable Hilbert space. J. Oper. Theory 42(2), 231–244 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chan, K.C., Taylor Jr., R.D.: Hypercyclic subspaces of a Banach space. Integr. Equ. Oper. Theory 41(4), 381–388 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Conway, J.B.: The Theory of Subnormal Operators. Mathematical Surveys and Monographs, vol. 36. American Mathematical Society, Providence (1991)Google Scholar
  12. 12.
    Conway, J.B.: A Course in Operator Theory. Graduate Studies in Mathematics, vol. 21. American Mathematical Society, Providence (2000)Google Scholar
  13. 13.
    Curto, R.E.: Spectral theory of elementary operators. In: Mathieu, M. (ed.) Elementary Operators & Applications (Blaubeuren, 1991), pp. 3–52. World Sci. Publ., New Jersey (1992)Google Scholar
  14. 14.
    Curto, R.E., Mathieu, M. (eds.): Elementary operators and their applications. In: Operator Theory: Advances and Applications, vol. 212. Birkhäuser, Basel (2011)Google Scholar
  15. 15.
    de la Rosa, M., Read, C.: A hypercyclic operator whose direct sum \(T\oplus T\) is not hypercyclic. J. Oper. Theory 61(2), 369–380 (2009)zbMATHGoogle Scholar
  16. 16.
    Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793–819 (1997)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Fialkow, L.A.: A note on the operator \(X\rightarrow AX-XB\). Trans. Am. Math. Soc. 243, 147–168 (1978)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fialkow, L.A.: A note on norm ideals and the operator \(X\longrightarrow AX-XB\). Israel J. Math. 32(4), 331–348 (1979)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fialkow, L.A.: A note on the range of the operator \(X\rightarrow AX-XB\). Ill. J. Math. 25(1), 112–124 (1981)zbMATHGoogle Scholar
  20. 20.
    Fialkow, L.A.: Structural properties of elementary operators. In: Mathieu, M. (ed.) Elementary Operators & Applications (Blaubeuren, 1991), pp. 55–113. World Sci. Publ., New Jersey (1992)Google Scholar
  21. 21.
    Gethner, R.M., Shapiro, J.H.: Universal vectors for operators on spaces of holomorphic functions. Proc. Am. Math. Soc. 100(2), 281–288 (1987)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gilmore, C., Saksman, E., Tylli, H.-O.: Hypercyclicity properties of commutator maps. Integr. Equ. Oper. Theory 87(1), 139–155 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Grivaux, S.: Hypercyclic operators with an infinite dimensional closed subspace of periodic points. Rev. Mat. Complut. 16(2), 383–390 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Grivaux, S.., Shkarin, S.: Non-mixing hypercyclic operators. Unpublished (2007)Google Scholar
  25. 25.
    Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear chaos (Universitext). Springer, London (2011)zbMATHGoogle Scholar
  26. 26.
    Gupta, M., Mundayadan, A.: Supercyclicity in spaces of operators. Results Math. 70(1), 95–107 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Halmos, P.R.: A Hilbert Space Problem Book, Volume 19 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1982)Google Scholar
  28. 28.
    Herrero, D.A.: Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99(1), 179–190 (1991)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kitai, C.: Invariant closed sets for linear operators. PhD thesis, University of Toronto (1982)Google Scholar
  30. 30.
    Kittaneh, F.: Normal derivations in norm ideals. Proc. Am. Math. Soc. 123(6), 1779–1785 (1995)MathSciNetzbMATHGoogle Scholar
  31. 31.
    León-Saavedra, F., Montes-Rodríguez, A.: Linear structure of hypercyclic vectors. J. Funct. Anal. 148(2), 524–545 (1997)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lumer, G., Rosenblum, M.: Linear operator equations. Proc. Am. Math. Soc. 10, 32–41 (1959)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Maher, P.J.: Commutator approximants. Proc. Am. Math. Soc. 115(4), 995–1000 (1992)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Martínez-Giménez, F., Peris, A.: Universality and chaos for tensor products of operators. J. Approx. Theory 124(1), 7–24 (2003)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Montes-Rodríguez, A., Romero-Moreno, M.C.: Supercyclicity in the operator algebra. Studia Math. 150(3), 201–213 (2002)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Petersson, H.: Hypercyclic conjugate operators. Integr. Equ. Oper. Theory 57(3), 413–423 (2007)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Rosenblum, M.: On the operator equation \(BX-XA=Q\). Duke Math. J. 23, 263–269 (1956)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer, London (2002)zbMATHGoogle Scholar
  39. 39.
    Saksman, E., Tylli, H.-O.: Multiplications and elementary operators in the Banach space setting. In: Castillo, J.M.F., Johnson, W.B. (eds.) Methods in Banach space theory, Volume 337 of London Math. Society of Lecture Note Series, pp. 253–292. Cambridge University Press, Cambridge (2006)Google Scholar
  40. 40.
    Salas, H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 993–1004 (1995)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Saldivia, L.E.: Topological transitivity of bounded linear operators. PhD thesis, Michigan State University (2003)Google Scholar
  42. 42.
    Tarbard, M.: Hereditarily indecomposable, separable \({\fancyscript {L}}_\infty \) Banach spaces with \(\ell _1\) dual having few but not very few operators. J. Lond. Math. Soc. (2) 85(3), 737–764 (2012)MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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