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arXiv · 0906.0160

Operator machines on directed graphs

Petr Hajek·Richard J. Smith
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Abstract

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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Petr Hajek, Richard J. Smith. 2009-05-31. Operator machines on directed graphs. https://doi.org/10.1007/s00020-010-1766-y

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