Failure of the wandering subspace property for analytic norm-increasing $3$-isometries

TL;DR

This paper constructs a specific example of an analytic norm-increasing 3-isometric weighted shift on a rootless directed tree that lacks the wandering subspace property, answering a longstanding question negatively.

Contribution

It provides the first explicit counterexample of a norm-increasing 3-isometric weighted shift without the wandering subspace property.

Findings

Counterexample on rootless quasi-Brownian directed tree

Negative answer to Shimorin's 2001 question

Illustrates limitations of the wandering subspace property for 3-isometries

Abstract

We construct an analytic norm-increasing $3$-isometric weighted shift on a rootless directed tree, which does not have the wandering subspace property. This answers a question of Shimorin [S2001, p. 185] in the negative. The counterexample in question is built over the rootless quasi-Brownian directed tree of valency $2.$