Operator machines on directed graphs
Petr Hajek, Richard J. Smith

TL;DR
The paper constructs a bounded linear operator on certain Banach spaces with symmetric bases, demonstrating the existence of a non-empty, nowhere dense set where the operator's iterates diverge, and showing convergence properties for points outside this set.
Contribution
It provides a counterexample to a recent conjecture by Prajitura, showing such operators exist on a broad class of Banach spaces with symmetric bases.
Findings
Existence of an operator with a non-empty, nowhere dense divergence set.
Points outside the divergence set have subsequences converging weakly to the original point.
The result applies to all classical Banach spaces containing an infinite-dimensional, complemented subspace with a symmetric basis.
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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Taxonomy
TopicsAdvanced Banach Space Theory · Holomorphic and Operator Theory · Advanced Operator Algebra Research
