Common non-trivial invariant closed cones for commuting contractions

Let T=(T1,…,TN) 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)∈σHe(T)⋂TN, where σHe(T) denotes the essential Harte spectrum of T and TN the unit politorus, respectively, then there exists a non-trivial cone C invariant for each contraction Tj;j∈{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].