Uniform exponential stability of periodic discrete switched linear system

Let {Πτ(m, n): m ≥ n ≥ 0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λτ(n), where τ:[0,1,2,⋯)→Ω is an arbitrary switching signal. We prove that the family {Πτ(m, n): m ≥ n ≥ 0} of bounded linear operator is uniformly exponentially stable if and only if the sequence n↦∑k=0neiαkΠτ(n,k)w(k):Z+→R is bounded.