Integral Equations and Operator Theory

, Volume 67, Issue 1, pp 15–31 | Cite as

Operator Machines on Directed Graphs



We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator \({R\,:\,X\longrightarrow\, X}\) such that the set
$$A = \{x \in X\,:\,{\left|\left|{R^n x}\right|\right|}\rightarrow \infty\}$$
is non-empty and nowhere norm-dense in X. Moreover, if \({x \in X\setminus A}\) then some subsequence of \({(R^n x)_{n=1}^\infty}\) converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all ‘classical’ Banach spaces admit such an operator.

Mathematics Subject Classification (2000)



Orbits of operators 


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  1. 1.
    Argyros, S.A., Haydon, R.G.: A hereditarily indecomposable \({\fancyscript{L}_\infty}\)-space that solves the scalar-plus-compact problem. Preprint.
  2. 2.
    Beauzamy B.: Introduction to operator theory and invariant subspaces. North-Holland Mathematical Library, vol. 42. North-Holland, Amsterdam (1988)Google Scholar
  3. 3.
    Caradus S.R.: Operators of Riesz type. Pac. J. Math 18, 61–71 (1966)MATHMathSciNetGoogle Scholar
  4. 4.
    Figiel T., Johnson W.B.: A uniformly convex Banach space which contains no p. Composito Math. 29, 179–190 (1977)MathSciNetGoogle Scholar
  5. 5.
    Jung I.B., Ko E., Pearcy C.: Some nonhypertransitive operators. Pac. J. Math 220, 329–340 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Maurey B.: Banach spaces with few operators. In: Johnson, W.B., Lindenstrauss, J. (eds) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1249–1297. Elsevier, Amsterdam (2003)Google Scholar
  7. 7.
    Müller V.: Orbits, weak orbits and local capacity of operators. Integral Equ. Oper. Theory 41, 230–253 (2001)CrossRefMATHGoogle Scholar
  8. 8.
    Müller, V.: Orbits of operators. In: Aizpuru-Tomás, A., León-Saavedra, F. (eds.) Advanced Courses of Mathematical Analysis I, pp. 53–79. Cádiz (2004)Google Scholar
  9. 9.
    Müller V., Vršovský J.: Orbits of linear operators tending to infinity. Rocky Mountain J. Math. 39, 219–230 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Prǎjiturǎ, G.: The geometry of an orbit. PreprintGoogle Scholar
  11. 11.
    Rolewicz S.: On orbits of elements. Studia Math. 32, 17–22 (1969)MATHMathSciNetGoogle Scholar
  12. 12.
    Tzafriri L.: On Banach spaces with unconditional bases. Israel J. Math. 17, 84–93 (1974)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute of Mathematics of the AS CRPraha 1Czech Republic
  2. 2.School of Mathematical SciencesUniversity College DublinBelfield, Dublin 4Ireland

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