Some new examples of universal hypercyclic operators in the sense of Glasner and Weiss

TL;DR

This paper introduces a simple criterion for identifying universal hypercyclic operators on Banach spaces, characterizes such operators among weighted shifts, and explores their existence and properties across various spaces.

Contribution

It provides a new criterion for universality, characterizes universal weighted shift operators, and studies the existence and necessary conditions of universal operators on different Banach spaces.

Findings

A general criterion for universality of operators.

Characterization of universal weighted shifts on nd 0.

Existence of universal operators on broad classes of Banach spaces.

Abstract

A bounded operator on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability space $(X,\mathcal {B},\mu )$, there exists an $A$-invariant probability measure $\nu $ on $H$ with full support such that the two dynamical systems $(X,\mathcal {B},\mu ;T)$ and $(H,\mathcal {B}_{H},\nu ;A)$ are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on $\ell_{p}$ or $c_{0}$, show the existence of universal operators on a large class of Banach spaces, and give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones), and study necessary conditions for an operator on a Hilbert space to be universal.