Universal elements for non-linear operators and their applications

TL;DR

This paper establishes conditions under which the direct sum of a universal operator and a multiplication operator remains universal, and applies this to characterize supercyclic operators and their cyclicity properties, answering open questions.

Contribution

It introduces topological conditions ensuring universality preservation under direct sums and characterizes supercyclic operators in new ways.

Findings

Direct sum of universal and multiplication operators can be universal under certain conditions.

Characterization of $ _+$-supercyclic operators.

Proof that certain direct sums of supercyclic operators are cyclic.

Abstract

We prove that under certain topological conditions on the set of universal elements of a continuous map $T$ acting on a topological space $X$, that the direct sum $T\oplus M_g$ is universal, where $M_g$ is multiplication by a generating element of a compact topological group. We use this result to characterize $\R_+$-supercyclic operators and to show that whenever $T$ is a supercyclic operator and $z_1,...,z_n$ are pairwise different non-zero complex numbers, then the operator $z_1T\oplus {...}\oplus z_n T$ is cyclic. The latter answers affirmatively a question of Bayart and Matheron.