Common non-trivial invariant closed cones for commuting contractions

Let T = (T-1,..., T-N) be a system of N commuting contractions defined on a infinite dimensional separable Hilbert space H. In this article, we will prove that if (1,..., 1) epsilon sigma H-e(T) boolean AND T-N, where sigma H-e(T) denotes the essential Harte spectrum of T and T-N the unit politorus, respectively, then there exists a nontrivial cone C invariant for each contraction T-j; j epsilon {1,..., N}. This result complements recent results of Tsatsomeros and co-workers [Roderick Edwards, Judith J. McDonald, Michael J. Tsatsomeros, On matrices with common invariant cones with applications in neural and gene networks, Linear Algebra Appl. 398 (2005) 37-67; Michael Tsatsomeros, A criterion for the existence of common invariant subspaces of matrices, Linear Algebra Appl. 322 (1-3) (2001) 51-59]. (C) 2008 Published by Elsevier Inc.