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Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 5, Pages 99–136 (Mi izv161)  

This article is cited in 49 scientific papers (total in 49 papers)

The generalized joint spectral radius. A geometric approach

V. Yu. Protasov

M. V. Lomonosov Moscow State University

Abstract: The properties of the joint spectral radius with an arbitrary exponent $p\in[1,+\infty]$ are investigated for a set of finite-dimensional linear operators $A_1,…,A_k$
\begin{align*} \widehat\rho_p&=\lim_{n\to\infty}(\dfrac{1}{k^n} \sum_\sigma\|A_{\sigma (1)}\cdots A_{\sigma(n)}\|^p)^{\frac{1}{pn}},\quad p<\infty,
\widehat\rho_{\infty}&=\lim_{n\to\infty}\max_{\sigma}\|A_{\sigma(1)}\cdots A_{\sigma(n)}\|^{\frac{1}{n}}, \end{align*}
where the summation and maximum extend over all maps
$$ \sigma \colon\{1,…,n\}\to\{1,…,k\}. $$

Using the operation of generalized addition of convex sets, we extend the Dranishnikov–Konyagin theorem on invariant convex bodies, which has hitherto been established only for the case $p=\infty$. The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius $\widehat \rho_p$. The problem of calculating $\widehat \rho_p$ for even integers $p$ is reduced to determining the usual spectral radius for an appropriate finite-dimensional operator. For other values of $p$, a geometric analogue of the method with a pre-assigned accuracy is constructed and its complexity is estimated.

DOI: https://doi.org/10.4213/im161

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English version:
Izvestiya: Mathematics, 1997, 61:5, 995–1030

Bibliographic databases:

MSC: 15A18, 90C60, 68Q25
Received: 28.05.1996

Citation: V. Yu. Protasov, “The generalized joint spectral radius. A geometric approach”, Izv. RAN. Ser. Mat., 61:5 (1997), 99–136; Izv. Math., 61:5 (1997), 995–1030

Citation in format AMSBIB
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