Integral Equations and Operator Theory

, Volume 67, Issue 1, pp 15–31

Operator Machines on Directed Graphs

Authors

  • Petr Hájek
    • Institute of Mathematics of the AS CR
    • School of Mathematical SciencesUniversity College Dublin
Article

DOI: 10.1007/s00020-010-1766-y

Cite this article as:
Hájek, P. & Smith, R.J. Integr. Equ. Oper. Theory (2010) 67: 15. doi:10.1007/s00020-010-1766-y

Abstract

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)

47A05

Keyword

Orbits of operators

Copyright information

© Birkhäuser / Springer Basel AG 2011