On the continuity of the generalized spectral radius in max algebra (Q541954)

The main result of this note is \(\mu (\Psi ) = \sup_{t\in (0,\infty)} (n^{-1}\rho (\Psi ^{(t)}))^{1/t}\), where \(\Psi\) is a set of \(n\times n\) bounded non-negative matrices, \(\rho (\Psi )\) and \(\mu (\Psi )\) denote respectively the generalized spectral radius of \(\Psi \) and its max version, and \(\Psi ^{(t)}\) denotes the Hadamard power of \(\Psi \). Further, this is a key result in a shorter proof of the known fact that \(\mu (\Psi )\) is continuous on the Hausdorff metric space \((\beta ,H)\) of all nonempty compact collections of \(n\times n\) non-negative matrices.