Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices

Let S = { S i } i ∈ I be an arbitrary family of complex n-by-n matrices, where 1 ⩽ n < ∞ . Let ρ ˆ ( S ) denote the joint spectral radius of S, defined as ρ ˆ ( S ) = lim sup ℓ → + ∞ { sup ( i 1 , … , i ℓ ) ∈ I ℓ ‖ S i 1 ⋯ S i ℓ ‖ 1 / ℓ } , which is independent of the norm ‖ ⋅ ‖ used here. A semi-norm ‖ ⋅ ‖ ⁎ on C n is called “extremal” of S, if it satisfies ‖ x ‖ ⁎ ≢ 0 and ‖ x ⋅ S i ‖ ⁎ ⩽ ρ ˆ ( S ) ‖ x ‖ ⁎ ∀ x = ( x 1 , … , x n ) ∈ C n and i ∈ I . In this paper, using an elementary analytical approach, we show that if S is bounded in C n × n , then there always exists, for S, an extremal semi-norm ‖ ⋅ ‖ ⁎ on C n ; if additionally S is compact in ( C n × n , ‖ ⋅ ‖ ) , this extremal semi-norm has the “Barabanov-type property”, i.e., to any x ∈ C n , one can find an infinite sequence i . : N → I with ‖ x ⋅ S i 1 ⋯ S i k ‖ ⁎ = ρ ˆ ( S ) k ‖ x ‖ ⁎ for each k ⩾ 1 . As a common starting point, this directly implies the fundamental results: Barabanovʼs Norm Theorem, Berger–Wangʼs Formula and Elsnerʼs Reduction Theorem.