Invariant ideals for uniform joint locally quasinilponent operators
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 \(T\)).
scientific article; zbMATH DE number 5614528