On the continuity of the generalized spectral radius in max algebra

Given a bounded set ΨΨ of n×nn×n non-negative matrices, let ρ(Ψ)ρ(Ψ) and μ(Ψ)μ(Ψ) denote the generalized spectral radius of ΨΨ and its max version, respectively. We show thatμ(Ψ)=supt∈(0,∞)n-1ρ(Ψ(t))1/t,whereΨ(t)Ψ(t) denotes the Hadamard power of ΨΨ. We apply this result to give a new short proof of a known fact that μ(Ψ)μ(Ψ) is continuous on the Hausdorff metric space (β,H)(β,H) of all nonempty compact collections of n×nn×n non-negative matrices.