关于infinite dimensional Volterra算子的Ω极限集
On omega limiting sets of infinite dimensional Volterra operators
摘要：
本文主要研究无穷维Volterra算子的极限行为。本文引入了两类无穷维Volterra算子$\tilde {\mathcal{V}}^+$和$\tilde{\mathcal{V}}^-$.对于从引入的类中选取的算子，我们分别研究了它们相对于$\ell^1$ -范数和逐点收敛的欧米茄极限集$ω_V$和$ω_V^{(w)}$。为了研究这些极限集之间的关系，我们研究了这类Volterra算子的线性李雅普诺夫函数。证明了：若Volterra算子属于$\tilde {\mathcal{V}}^+$，则对每个$\xb\in S$，其集合与$ω_V^{(w)}(\xb)$重合，且非空.若Volterra算子属于$\tilde {\mathcal{V}}^-$，则$ω_V(\xb)$可以为空，并暗示$V$的非遍历性（关于$\ell^1$ -范数），但它是弱遍历的.
英文摘要：
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 $ω_V$ and $ω_V^{(w)}$ 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 $ω_V^{(w)}(\xb)$ coincide for every $\xb\in S$, and moreover, they are non empty. If Volterra operator belongs to $\tilde {\mathcal{V}}^-$, then $ω_V(\xb)$ could be empty, and it implies the non-ergodicity (w.r.t $\ell^1$-norm) of $V$, while it is weak ergodic.
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