A note on a gap result for norms of semigroups of matrices (Q865401)

Let \(\| \cdot \| \) be a matrix norm and \(\Sigma \) be a bounded set of \(n\times n\) complex matrices. For \(m\geq 1,\) let \(\Sigma _m\) be the set of all products of length \(m\) of matrices in \(\Sigma .\) Let \(S(\Sigma )\) denote the multiplicative semigroup generated by \(\Sigma .\) The generalized spectral radius of \(\Sigma,\) \(\rho (\Sigma ),\) is defined by \(\rho (\Sigma )=\limsup_{m\rightarrow \infty } [\rho_m(\Sigma)]^{\frac{1}{m}},\) where \(\rho_m(\Sigma)=\sup _{A\in \Sigma_m}\rho(A),\) and \(\rho (A)\) is the spectral radius of \(A.\) The joint spectral radius of \(\Sigma,\) \(\hat{\rho} (\Sigma ),\) is defined by \(\hat{\rho} (\Sigma )=\limsup_{m\rightarrow \infty } [\hat{\rho} _m(\Sigma ,\| \cdot \| )]^{\frac{1}{m}},\) where \(\hat{\rho} _m(\Sigma, \| \cdot \| )=\sup _{A\in \Sigma _m}\| A\| .\) The authors prove that either there is some constant \(\alpha >1\) such that \(\hat{\rho} _m(\Sigma, \| \cdot \| )>\alpha ^m\) for all \(m\) sufficiently large, or \(\hat{\rho} _m(\Sigma, \| \cdot \| )=O(m^{n-1}).\) Moreover, \(\hat{\rho} _m(\Sigma, \| \cdot \| )=O(m^{n-1})\) if and only if the eigenvalues of every matrix in the semigroup generated by \(\Sigma\) are all on or inside the unit circle.