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

Let S={Si}i∈IS={Si}i∈I be an arbitrary family of complex n-by-n   matrices, where 1⩽n<∞1⩽n<∞. Let ρˆ(S) denote the joint spectral radius of S, defined asρˆ(S)=lim supℓ→+∞{sup(i1,…,iℓ)∈Iℓ‖Si1⋯Siℓ‖1/ℓ}, which is independent of the norm ‖⋅‖‖⋅‖ used here. A semi-norm ‖⋅‖⁎‖⋅‖⁎ on CnCn is called “extremal” of S, if it satisfies‖x‖⁎≢0and‖x⋅Si‖⁎⩽ρˆ(S)‖x‖⁎∀x=(x1,…,xn)∈Cn and i∈I. In this paper, using an elementary analytical approach, we show that if S   is bounded in Cn×nCn×n, then there always exists, for S  , an extremal semi-norm ‖⋅‖⁎‖⋅‖⁎ on CnCn; if additionally S   is compact in (Cn×n,‖⋅‖)(Cn×n,‖⋅‖), this extremal semi-norm has the “Barabanov-type property”, i.e., to any x∈Cnx∈Cn, one can find an infinite sequence i.:N→Ii.:N→I with ‖x⋅Si1⋯Sik‖⁎=ρˆ(S)k‖x‖⁎ for each k⩾1k⩾1. As a common starting point, this directly implies the fundamental results: Barabanovʼs Norm Theorem, Berger–Wangʼs Formula and Elsnerʼs Reduction Theorem.