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 $xbin S$, 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.
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