Is there any nontrivial compact generalized shift operator on Hilbert spaces?

  • Fatemah Ayatollah Zadeh ShiraziEmail author
  • Fatemeh Ebrahimifar


In the following text for cardinal number \(\tau >0\), and self-map \(\varphi :\tau \rightarrow \tau \) we show the generalized shift operator \(\sigma _\varphi (\ell ^2(\tau ))\subseteq \ell ^2(\tau )\) (where \(\sigma _\varphi ((x_\alpha )_{\alpha<\tau })=(x_{\varphi (\alpha )})_{\alpha <\tau }\) for \((x_\alpha )_{\alpha <\tau }\in {{\mathbb {C}}}^\tau \)) if and only if \(\varphi :\tau \rightarrow \tau \) is bounded and in this case Open image in new window is continuous, consequently Open image in new window is a compact operator if and only if \(\tau \) is finite.


Compact operator Generalized shift Hilbert space 

Mathematics Subject Classification



  1. 1.
    Ayatollah Zadeh Shirazi, F., Dikranjan, D.: Set theoretical entropy: a tool to compute topological entropy. In: Proceedings ICTA 2011, Islamabad, Pakistan, July 4–10, 2011, pp. 11–32. Cambridge Scientific Publishers (2012)Google Scholar
  2. 2.
    Ayatollah Zadeh Shirazi, F., Karami Kabir, N., Heidari Ardi, F.: A note on shift theory, Mathematica Pannonica. In: Proceedings of ITES-2007, vol. 19/2, pp. 187–195 (2008)Google Scholar
  3. 3.
    Conway, J.B.: A Course in Abstract Analysis, Graduate Studies in Mathematics, vol. 141. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  4. 4.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics, 2nd edn. A Wiley-Interscience Publication, New York (1999)zbMATHGoogle Scholar
  5. 5.
    Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos, Universitext. Springer, Berlin (2011)CrossRefGoogle Scholar
  6. 6.
    Giordano Bruno, A.: Algebraic entropy of generalized shifts on direct products. Commun. Algebra 38(11), 4155–4174 (2010)CrossRefGoogle Scholar
  7. 7.
    Holz, M., Steffens, K., Weitz, E.: Introduction to Cardinal Arithmetic, Birkhuser Advanced Texts: Basler Lehrbcher. Birkhuser Verlag, Basel (1999)CrossRefGoogle Scholar
  8. 8.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley Classics Library, New York (1989)zbMATHGoogle Scholar
  9. 9.
    Ornstein, D.: Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337–352 (1970)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79. Springer, Berlin (1982)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  • Fatemah Ayatollah Zadeh Shirazi
    • 1
    Email author
  • Fatemeh Ebrahimifar
    • 1
  1. 1.Faculty of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

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