Asymptotic Stability and Generalized Gelfand Spectral Radius Formula

Let ∑ be a set of n × n complex matrices. For m = 1, 2, …, let ∑m be the set of all products of matrices in ∑ of length m. Denote by ∑′ the multiplicative semigroup generated by ∑. ∑ is said to be asymptotically stable (in the sense of dynamical systems) if there is 0 < α < 1 such that there are bounded neighborhoods U, V ⊂ Cn of the origin for which AV ⊂ αmU for all A ∈ ∑m, m = 1, 2, …. For a bounded set ∑ of n × n complex matrices, it is shown that the following conditions are mutually equivalent: (i) ∑ is asymptotically stable; (ii) \g9(∑) = lim supm → ∞[supA ∈ ∑m ‖ A ‖]1/m < 1; (iii) ϱ(∑) = lim supm → ∞[supA ∈ ∑m ϱ(A)]1/m < 1, where ϱ(A) stands for the spectral radius of A; and (iv) there exists a positive number α such that ϱ(A) ⩽ α < 1 for all A ∈ ∑′. This fact answers an open question raised by Brayton and Tong. The generalized Gelfand spectral radius formula, that is, ϱ(∑) = \g9(∑), conjectured by Daubechies and Lagarias and proved by Berger and Wang using advanced tools from ring theory and then by Elsner using analytic-geometric tools, follows immediately form the above asymptotic stability theorem.