On the continuity of the generalized spectral radius in max algebra

Abstract

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