Simultaneous Schur stability

Let Sigma be a set of n x n complex matrices. Denote by L(Sigma) the multip licative semigroup generated by Sigma. For an n x n complex matrix A, the s pectrum of A and spectral radius of A are denoted by a(A) and r(A), respect ively. Motivated by a phenomenon in the closed unit disc of the complex pla ne, we give a rigorous definition of simultaneous Schur stability as follow s. Sigma is said to be simultaneously Schur stable if r(A) less than or equ al to 1 (A is an element of L(Sigma)) and 1 is not an element of sigma(A) ( A is an element of L(Sigma)). Sigma is said to be simultaneously Schur stab le if there exists a norm parallel to . parallel to on C-n such that sup{pa rallel to A parallel to;A is an element of Sigma} < 1. It is proved that fo r a bounded set Sigma of n x n complex matrices, Sigma is asymptotically st able if and only if it is simultaneously Schur stable. By way of "simultane ous Schur stability", some applications are illustrated, especially an anal ytic-combinatorial proof of a recent result of considerable depth: Generali zed Gelfand spectral radius formula. (C) 1999 Published by Elsevier Science Inc. All rights reserved.