Invariant Subspaces for Positive Operators Acting on a Banach Space with Basis

Abstract

Recently we established several invariant subspace theorems for operators acting on an ${l_p}$-space. In this note we extend these results from operators acting on an ${l_p}$-space to operators acting on any Banach space with a (not necessarily unconditional) Schauder basis. For instance, it is shown that if a continuous quasinilpotent operator on a Banach space is positive with respect to the closed cone generated by a basis, then the operator has a nontrivial closed invariant subspace.