Bounds on the joint and generalized spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

Let Ψ1,…,Ψm be bounded sets of positive kernel operators on a Banach function space L. We prove that for the generalized spectral radius ρ and the joint spectral radius ρˆ the inequalitiesρ(Ψ1(1m)∘⋯∘Ψm(1m))≤ρ(Ψ1Ψ2⋯Ψm)1m,ρˆ(Ψ1(1m)∘⋯∘Ψm(1m))≤ρˆ(Ψ1Ψ2⋯Ψm)1m hold, where Ψ1(1m)∘⋯∘Ψm(1m) denotes the Hadamard (Schur) geometric mean of the sets Ψ1,…,Ψm.