Infinite matrices with “few” non-zero entries and without non-trivial invariant subspaces

In this paper we investigate a family of infinite matrices that act on ℓ1. We derive a condition sufficient to guarantee that a matrix has no non-trivial closed invariant subspaces. As a result, a simplest known operator on ℓ1 without invariant subspaces is obtained. All entries of the matrix of the example but one are non-negative.