A mixing operator $T$ for which $(T, T^2)$ is not disjoint transitive

TL;DR

This paper constructs a specific mixing operator on a Hilbert space demonstrating that the pair $(T, T^2)$ does not exhibit disjoint transitivity, answering a question in ergodic theory.

Contribution

It introduces a novel mixing operator where the tuple $(T, T^2)$ is not disjoint transitive, expanding understanding of operator dynamics.

Findings

Constructed a mixing operator with non-disjoint transitive $(T, T^2)$

Answered an open question in ergodic Ramsey theory

Shows limitations of mixing properties in operator tuples

Abstract

Using a result from ergodic Ramsey theory, we answer a question posed by B\`es, Martin, Peris and Shkarin by showing a mixing operator $T$ on a Hilbert space such that the tuple $(T, T^2)$ is not disjoint transitive.