Operator machines on directed graphs

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.