A new sufficient condition for the uniqueness of Barabanov norms (Q2910957)

The joint spectral radius \(\rho({\mathcal A})\) of a bounded set \({\mathcal A}\) of \(d\)-by-\(d\) matrices over \(\mathbb{K}\) (\(=\mathbb{R}\) or \(\mathbb{C}\)) is defined by \textit{G.-C. Rota} and \textit{W. G. Strang} [Nederl. Akad. Wet., Proc., Ser. A 63, 379--381 (1960; Zbl 0095.09701)], to be \(\limsup_{n\to\infty}\{\| A_{i_1}\cdots A_{i_n}\|^{1/n}: A_{i_j}\in{\mathcal A}\}\). \({\mathcal A}\) is said to be irreducible if there is no proper subspace of \(\mathbb{K}^d\) which is left invariant under all elements of \({\mathcal A}\). \textit{N. E. Barabanov} showed [in 1988] that, for any compact irreducible set \({\mathcal A}\) of \(M_d(\mathbb{K})\), there is associated a norm \(|||.|||\) on \(\mathbb{K}^d\) such that \(\rho({\mathcal A})|||v|||= \sup\{|||Av|||: A\in{\mathcal A}\}\) holds for all vectors \(v\) in \(\mathbb{K}^d\). The present paper is concerned with a sufficient condition for the uniqueness of such a Barabanov norm for \({\mathcal A}\).NEWLINENEWLINE The main result, Theorem 2.1, says that if \({\mathcal A}\) is a nonempty bounded irreducible subset of \(M_d(\mathbb{K})\) such that the limit semigroup \(\&({\mathcal A})\equiv\bigcap^\infty_{m=1}\, (\overline{\bigcup^\infty_{n=m} \rho(A)^{-n}{\mathcal A}^n})\), where \({\mathcal A}^n= \{A_{i_1}\cdots A_{i_n}: A_{i_j}\in{\mathcal A}\}\), is such that for every pair of nonzero vectors \(v_1\) and \(v_2\) in \(\mathbb{K}^d\), there exist \(B_1\) and \(B_2\) in \(\&({\mathcal A})\) and \(\lambda\) in \(\mathbb{K}\) such that \(B_1v_1=\lambda v_2\) and \(B_2v_2= \lambda^{-1} v_1\), then the Barabanov norm for \({\mathcal A}\) is unique up to a positive scalar constant. Examples of \({\mathcal A}\) consisting of 2-by-2 real matrices are given to illustrate this condition and to contrast with the ones previously obtained by the author. A theoretical application of the main theorem is also given, which shows that there is a nonempty open set \(\nu\) of ordered pairs of 2-by-2 real matrices such that the sets \(\nu_1\) (resp., \(\nu_2\)) consisting of elements in \(\nu\) which have (resp., do not have) a unique Barabanov norm are both dense in \(\nu\). This is in contrast to a previous result of the author that there is a nonempty open set \(u\) of pairs of 2-by-2 real matrices for which every pair in \(u\) has a unique Barabanov norm.