Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on lp-Spaces

Abstract

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 \({\mathcal{C}} \neq \{0\}\) is a collection of continuous operators with moduli on lp that is finitely modulus-quasinilpotent at a non-zero positive vector x0 then \({\mathcal{C}}\) and its right modulus sub-commutant \({\mathcal{C}}_{m}^{\prime}\) have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on lp whose moduli commute with a non-zero 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.