There seems to be many valid ways of generalizing the notion of the spectral radius $\rho(A)$ of a complex matrix $A$ to spectral radii of multiple operators. I am wondering if there is an abstract theory of what it means to be a multi-spectral radius $\rho(A_1,\dots,A_r)$ of complex matrices $A_1,\dots,A_r$.

Example 0: Suppose that $A_1,\dots,A_r$ are complex matrices and $1\leq p<\infty$. Then define $$\rho_p(A_1,\dots,A_r)=\lim_{n\rightarrow\infty}\big(\sum_{i_1,\dots,i_n\in\{1,\dots,r\}}\|A_{i_1}\dots A_{i_r}\|^p\big)^{1/(pn)}.$$

We can call this notion of the spectral radius the $L_p$-spectral radius.

Theorem: $\rho_2(A_1,\dots,A_r)^2=\rho(A_1\otimes\overline{A_1}+\dots+A_r\otimes\overline{A_r})$. Here, $\overline{A}=(A^*)^T=(A^T)^*$. Alternatively, define the completely positive linear mapping $\Phi(A_1,\dots,A_r):M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ by setting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+A_rXA_r^*$. Then $\rho_2(A_1,\dots,A_r)^2=\rho(\Phi(A_1,\dots,A_r))$.

It seems like $\rho_2(A_1,\dots,A_r)$ is the best way of generalizing the notion of a spectral radius to multiple operators if I had to choose one notion of a spectral radius.

Example 1: Let $\mathfrak{k}=(k_n)_{n=0}^{\infty}$ be a sequence of numbers in the set $\{1,\dots,r\}$. Then define $\rho_\mathfrak{k}(A_1,\dots,A_r)=\limsup_{n\rightarrow\infty}\|A_{k_0}\dots A_{k_n}\|^{1/n}.$

Example 2: Define $\rho_{2,1}(A_1,\dots,A_r)$ to be the largest value of $(|z_1^2|+\dots+|z_r^2|)^{-1/2}$ where $I-(z_1A_1+\dots+z_rA_r)$ is not invertible. Then $\rho_{2,1}$ is another generalized notion of a spectral radius. More generally, if $\|\cdot\|$ is a complex norm on $\mathbb{C}^n$, then define $\rho_{2,\|\cdot\|}(A_1,\dots,A_r)$ to be the largest value of $\|(z_1,\dots,z_r)\|^{-1}$ where $I-(z_1A_1+\dots+z_rA_r)$ is not invertible.

If $\rho$ is a multi-spectral radius function, then it seems like $\rho$ should satisfy properties such as log-plurisubharmonicity, continuity, homogeneity of degree 1, invariance under joint-similarity, and a few mundane properties. For some particularly nice multi-spectral radius functions $\rho$, the value $\rho(A_1,\dots,A_r)$ only depends on the completely positive superoperator $\Phi(A_1,\dots,A_r)$. And if a multi-spectral radius function $\rho$ only depends on the superoperator $\Phi(A_1,\dots,A_r)$, then I would imagine that this multi-spectral radius easily generalizes to bounded operators between Hilbert spaces.

What would be a good axiomatization for multi-spectral radius functions? Is there an axiomatization for what is meant by a multi-spectrum of multiple operators? Clearly, one may consider the spectrum of $\Phi(A_1,\dots,A_r)$ as a multi-spectrum of $A_1,\dots,A_r$. However, if $z_1,\dots,z_r$ are complex numbers with $|z_1|^2+\dots+|z_r|^2=1$ and where $\rho(z_1A_1+\dots+z_rA_r)$ is maximized, then the spectrum $\sigma(z_1A_1+\dots+z_rA_r)$ may be considered as another notion of a multi-spectrum.

I am interested in these generalized spectral radii since I have originally used a generalized notion of the spectral radius for cryptocurrency research and development, but such generalized spectral radii seem applicable for other machine learning applications.