The generalized spectral-radius theorem: An analytic-geometric proof Original Research Article

Let ∑ be a bounded set of complex matrices,∑m = {A1 … Am: Ai set membership, variant ∑}. The generalized spectral-radius theorem states thatvarrho(∑) =ρ?(∑), where varrho(∑) and ρ?(σ) are defined as follows:varrho{∑) =lim supvarrhom(∑){1/m}, wherevarrhom(∑) =sup{varrho(A): A set membership, variant ∑m} with varrho (A) the spectral radius;ρ?(∑) =lim supρ?m(∑){1/m}, whereρ?m(∑) =sup{double vertical barAdouble vertical bar: A set membership, variant ∑m} with double vertical bar double vertical bar any matrix norm. We give an elementary proof, based on analytic and geometric tools, which is in some ways simpler than the first proof by Berger and Wang.