In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an lp-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on lp that is quasinilpotent at a non-zero positive vector x0 has a non-trivial invariant closed subspace. In this paper, it is proved that if C = {0} is a collection of continuous operators with moduli on lp that is finitely modulus-quasinilpotent at a non-zero positive vector x0 then C and its right modulus sub-commutant C m have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on lp whose moduli commute with a nonzero positive operator I on lp that is quasinilpotent at a non-zero positive vector x0 have a common non-trivial invariant closed subspace, so that all positive operators on lp which commute with a non-zero positive operator S on lp that is quasinilpotent at a non-zero positive vector x0 have a common non-trivial invariant closed subspace.
Mathematics Subject Classification (2000). 47A15, 47B65, 46B40.