The joint spectral radius and invariant sets of the several linear operators

V. Yu. Protasov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 1, pp. 205-231.

Résumé

This paper concerns the properties of the joint spectral radius of the several linear $n$-dimensional operators: $$ \hat{\rho}(A_1,\ldots,A_k)=\lim\limits_{m\to\infty}\,\max\limits_{\sigma} \|A_{\sigma (1)}\ldots A_{\sigma (m)}\|^{\frac{1}{m}},\quad \sigma\colon\ \{1,\ldots,m\}\to\{1,\ldots,k\}. $$ The theorem of Dranishnikov–Konyagin on the existence of invariant convex set $M$ for several linear operators is proved. $\operatorname{Conv}(A_1M,\ldots,A_kM)=\lambda M$, $\lambda=\hat{\rho}(A_1,\ldots,A_k)$. Paper concludes with several boundary propositions on construction of the invariant sets, some properties of the invariant sets and algorithm of finding the joint spectral radius with estimation of its difficulty.