# Operator machines on directed graphs

Petr Hajek, Richard J. Smith

Arxiv ID: 0906.0160•Last updated: 6/14/2022

We show that if an infinite-dimensional Banach space X has a symmetric basis
then there exists a bounded, linear operator R : X --> X such that the set
A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X.
Moreover, if x in X\A then some subsequence of (R^n(x)) converges weakly to x.
This answers in the negative a recent conjecture of Prajitura. 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.

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