On omega limiting sets of infinite dimensional Volterra operators

TL;DR

This paper investigates the long-term behavior of infinite dimensional Volterra operators by classifying them into two types and analyzing their omega limiting sets under different convergence modes, revealing conditions for ergodicity.

Contribution

It introduces two classes of infinite dimensional Volterra operators and studies their omega limiting sets, establishing relations between these sets and conditions for ergodicity.

Findings

For operators in fVf^+, the omega sets coincide and are non-empty.

Operators in fVf^- may have empty omega sets, indicating non-ergodicity.

The study links the structure of Volterra operators to their ergodic properties via limiting sets.

Abstract

In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $\tilde {\mathcal{V}}^+$ and $\tilde{\mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $\omega_V$ and $\omega_V^{(w)}$ with respect to $\ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $\tilde {\mathcal{V}}^+$, then the sets and $\omega_V^{(w)}(\xb)$ coincide for every $\xb\in S$, and moreover, they are non empty. If Volterra operator belongs to $\tilde {\mathcal{V}}^-$, then $\omega_V(\xb)$ could be empty, and it implies the non-ergodicity (w.r.t $\ell^1$-norm) of $V$, while it is weak ergodic.