Operator machines on directed graphs
Functional Analysis
2022-06-14 v1
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.
Cite
@article{arxiv.0906.0160,
title = {Operator machines on directed graphs},
author = {Petr Hajek and Richard J. Smith},
journal= {arXiv preprint arXiv:0906.0160},
year = {2022}
}