Invariant ideals for uniform joint locally quasinilponent operators (Q734589)

It is the aim of the paper under review to present some results regarding the invariant subspace problem associated to an \(N\)-tuple \(T=(T_1,\dots,T_N)\) of positive linear operators acting on a Banach lattice \(E\). These results extend, to the case of several variables, certain theorems due to \textit{Y.\,A.\thinspace Abramovich, C.\,D.\thinspace Aliprantis} and \textit{O.\,Burkinshaw} [J.~Funct.\ Anal.\ 124, No.~1, 95--111 (1994; Zbl 0819.47049)] and \textit{Y.\,A.\thinspace Abramovich} and \textit{C.\,D.\thinspace Aliprantis} [``An invitation to operator theory'' (Graduate Studies in Mathematics 50; Providence/RI:\ AMS) (2002; Zbl 1022.47001)]. It is shown, for example, that \(T\) has a common nontrivial closed invariant ideal if there exists a positive operator \(S\) on \(E\) which is locally quasinilpotent at some \(x_0>0\), dominates a nonzero compact operator and satisfies \(ST_j\leq T_jS\;(j=1,\dots,N)\). A similar conclusion holds true if \(T\) is joint compact-friendly (i.e., there exist three nonzero operators \(R,K\) and \(A\) on \(E\) with \(R,K\) positive and \(K\) compact such that \(R\) commutes with each \(T_j\) and \(A\) is dominated by both \(R\) and \(K\)) and uniform joint locally quasinilpotent at some \(x_0>0\) (i.e., \(\lim_{n\to\infty}\max_{S\in T^n}\|Sx_0\|^{1/n}=0\), where \(T^n\) denotes the set of all possible products of \(n\) elements in \(T\)).