ℓ∞ℓ∞-stability analysis of discrete autonomous systems described by Laurent polynomial matrix operators

In this paper, we analyze the ℓ∞ℓ∞-stability of infinite dimensional discrete autonomous systems, whose dynamics is governed by a Laurent polynomial matrix A(σ,σ−1)A(σ,σ−1) in shift operator σσ on vector valued sequences. We give necessary and sufficient conditions for the ℓ∞ℓ∞-stability of such systems. We also give easy to check tests to conclude or to rule out the ℓ∞ℓ∞-stability of such systems.