, Volume 67, Issue 1, pp 15-31
Date: 20 Mar 2010

Operator Machines on Directed Graphs

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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.

Both authors are supported by Grant A 100190801 and Institutional Research Plan AV0Z10190503.