Generalized spectral radius and its max algebra version

Let Sigma subset of C-nxn and Psi subset of R-+(nxn) likra be bounded subsets and let rho(Sigma) and mu(Psi) denote the generalized spectral radius of Sigma and the max algebra version of the generalized spectral radius of Psi, respectively. We apply a single matrix description of mu(Psi) to give a new elementary and straightforward proof of the Berger-Wang formula in max algebra and consequently a new short proof of the original Berger-Wang formula in the case of bounded subsets of n x n non-negative matrices. We also obtain a new description of mu(Psi) in terms of the Schur-Hadamard product and prove new trace and max-trace descriptions of mu(Psi) and rho(Sigma). In particular, we show that mu(Psi) = lim(m ->infinity)sup [sup(A is an element of Psi circle times m) tr(circle times)(A)](1/m) = lim(m ->infinity)sup [sup(A is an element of Psi circle times m) tr(A)](1/m) and rho(Sigma) = lim(m ->infinity)sup [sup(B is an element of Sigma m) tr vertical bar B vertical bar](1/m) = lim(m ->infinity)sup [sup(B is an element of Sigma m) tr(circle times)vertical bar B vertical bar](1/m,) where tr(circle times)(A) = max(i=1, ... ,n) a(ii) and vertical bar B vertical bar =[vertical bar b(ij)vertical bar]. (C) 2012 Elsevier Inc. All rights reserved.