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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
$A_1,\dots,A_k$
\begin{align*}
\widehat\rho_p&=\lim_{n\to\infty}\biggl(\dfrac{1}{k^n}\,\sum_\sigma\|A_{\sigma (1)}\cdots A_{\sigma(n)}\|^p\biggr)^{\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
$$
\sigma \colon\{1,\dots,n\}\to\{1,\dots,k\}.
$$
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.
MSC: 15A18,
90C60,
68Q25 Received: 28.05.1996
DOI:
10.4213/im161
Fulltext:
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