The generalized joint spectral radius. A~geometric approach

V. Yu. Protasov

Izvestiya. Mathematics , Tome 61 (1997) no. 5, pp. 995-1030

Résumé

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.