Generalized spectral radius and its max algebra version

Let Σ⊂Cn×nΣ⊂Cn×n and Ψ⊂R+n×n be bounded subsets and let ρ(Σ)ρ(Σ) and μ(Ψ)μ(Ψ) denote the generalized spectral radius of ΣΣ and the max algebra version of the generalized spectral radius of ΨΨ, respectively. We apply a single matrix description of μ(Ψ)μ(Ψ) 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×nn×n non-negative matrices. We also obtain a new description of μ(Ψ)μ(Ψ) in terms of the Schur–Hadamard product and prove new trace and max-trace descriptions of μ(Ψ)μ(Ψ) and ρ(Σ)ρ(Σ). In particular, we show thatμ(Ψ)=limsupm→∞supA∈Ψ⊗mtr⊗(A)]1/m=limsupm→∞[supA∈Ψ⊗mtr(A)]1/mandρ(Σ)=limsupm→∞[supB∈Σmtr(|B|)]1/m=limsupm→∞[supB∈Σmtr⊗(|B|)]1/m,where tr⊗(A)=maxi=1,…,naiitr⊗(A)=maxi=1,…,naii and |B|=[|bij|]|B|=[|bij|].