Asymptotic stability and generalized Gelfand spectral radius formula Original Research Article

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 subset of Cn of the origin for which AV subset of αmU for all A set membership, variant ∑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) image; (iii) varrho(∑) = lim supm → ∞[supA set membership, variant ∑m varrho(A)]1/m < 1, where varrho(A) stands for the spectral radius of A; and (iv) there exists a positive number α such that varrho(A) less-than-or-equals, slant α < 1 for all A set membership, variant ∑′. This fact answers an open question raised by Brayton and Tong. The generalized Gelfand spectral radius formula, that is, image, 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.