Hyperinvariant Closed Ideals for a Finitely Quasinilpotent Collection of Operators

In this paper, it is proved that if $\mathscr C\ne\{0\}$ is a collection of continuous operators with modulus on an $\ell_p$-space ($1\le p<\infty$) that is finitely modulus-quasinilpotent at a nonzero positive vector $x_0$ in $\ell_p$, then $\mathscr C$ and its right modulus sub-commutant $\mathscr C'_m$ have a common nontrivial invariant closed ideal.