An example of a minimal action of the free semi-group $\F^{+}_{2}$ on the Hilbert space

TL;DR

This paper constructs a specific bounded linear operator on an infinite-dimensional Hilbert space demonstrating a minimal action of the free semi-group with two generators, addressing the invariant subset problem.

Contribution

It introduces a new example of a minimal semi-group action on a Hilbert space, linking the invariant subset problem to semi-group dynamics.

Findings

Existence of a bounded operator with dense orbits for vectors or their images.

Construction of a minimal action of the free semi-group on a Hilbert space.

Connection between dense orbits and semi-group actions in operator theory.

Abstract

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator $T$ on a separable infinite-dimensional Hilbert space $H$ such that the orbit $\{T^{n}x;\ n\ge 0\}$ of every non-zero vector $x\in H$ under the action of $T$ is dense in $H$. We show that there exists a bounded linear operator $T$ on a complex separable infinite-dimensional Hilbert space $H$ and a unitary operator $V$ on $H$, such that the following property holds true: for every non-zero vector $x\in H$, either $x$ or $Vx$ has a dense orbit under the action of $T$. As a consequence, we obtain in particular that there exists a minimal action of the free semi-group with two generators $\F^{+}_{2}$ on a complex separable infinite-dimensional Hilbert space $H$.