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Fundamentalnaya i Prikladnaya Matematika, 1996, Volume 2, Issue 1, Pages 205–231
(Mi fpm141)
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This article is cited in 59 scientific papers (total in 60 papers)
The joint spectral radius and invariant sets of the several linear operators
V. Yu. Protasov M. V. Lomonosov Moscow State University
Abstract:
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.
Received: 01.03.1995
Citation:
V. Yu. Protasov, “The joint spectral radius and invariant sets of the several linear operators”, Fundam. Prikl. Mat., 2:1 (1996), 205–231
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