Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on \(l_{ p}\)-spaces (Q2373349)

This article deals with the work initiated in [J.~Funct.\ Anal.\ 115, No.\,2, 418--424 (1993; Zbl 0819.47006)] by \textit{Yu.\,A.\thinspace Abramovich}, \textit{C.\,D.\thinspace Aliprantis} and \textit{O.\,Burkinshaw}. Let \(X\) be a Banach space and let \(B(X)\) be the Banach algebra of all bounded linear operators from \(X\) into itself. Let \(\mathcal C\neq\{0\}\subset B(X)\) be a collection of positive operators on an \(l_{p}\)-space \(X\) (\(1\leq p<\infty\)) that is finitely quasinilpotent at some nonzero positive vector \(x_0\in X\), i.e., \(r({\mathcal F},x_0)=0\) for any finite subset \({\mathcal F}\) of \(\mathcal C\), where \(r({\mathcal F},X_0)=\lim\sup_{n\to\infty}\|{\mathcal F}^n(x_0)\|^{(1/n)}\) denotes the joint spectral radius of \({\mathcal F}\). The authors show that \(\mathcal C\) and the the right modulus subcommutant \(\mathcal C_{m}'\) have a common nontrivial invariant closed subspace. In particular, all continuous operators with moduli on \(l_{p}\) whose moduli commute with a nonzero positive operator on \(l_{p}\) that is quasinilpotent at a nonzero positive vector \(x_0\) have a common nontrivial invariant closed subspace.