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.

中文翻译：

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关于无穷维 Volterra 算子的 omega 限制集

在本文中，我们的目标是研究无限维 Volterra 算子的极限行为。我们介绍了两个类 $\tilde {\mathcal{V}}^+$ 和 $\tilde{\mathcal{V}}^-$ 的无限维 Volterra 算子。对于从引入的类中提取的算子，我们分别研究了它们关于 $\ell^1$-norm 和逐点收敛的 omega 限制集 $\omega_V$ 和 $\omega_V^{(w)}$。为了研究这些限制集之间的关系，我们研究了此类 Volterra 算子的线性李雅普诺夫函数。证明如果 Volterra 算子属于 $\tilde {\mathcal{V}}^+$，那么对于每一个 $\xb\in S，集合和 $\omega_V^{(w)}(\xb)$ 重合$，而且，它们不是空的。如果 Volterra 算子属于 $\tilde {\mathcal{V}}^-$，则 $\omega_V(\xb)$ 可能为空，这意味着非遍历性（wr