Stability of discrete linear inclusion

Let M = Ai be a set of linear operators on n. The discrete linear inclusion DLI(M) is the set of possible trajectories (xi: i ⩾ 0) such that xn = AinAin−1 … Ai1x0 where and Aij ∈ M. We study several notions of stability for DLI(M), including absolute asymptotic stability (AAS), which is that all products Ain … Ai1 → 0 as n → ∞. We mainly study the case that M is a finite set. We give criteria for the various forms of stability. Two new approaches are taken: one relates the question of AAS of DLI(M) to formal language theory and finite automata, while the second connects the AAS property to the structure of a Lie algebra associated to the elements of M. More generally, the discrete linear inclusion DLI(M) makes sense for M contained in a Banach algebra . We prove some results for AAS in this case, and give counterexamples showing that some results valid for finite sets of operators on n are not true for finite sets M in a general Banach algebras .