Suppose that $X$ is a finite dimensional Hilbert space and $A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be $$\limsup_{n\rightarrow\infty}(\sum_{a_{1},\dots,a_{n}\in\{1,\dots,r\}}\|A_{a_{1}}\dots A_{a_{n}}\|)^{1/n}.$$ It is not too hard to convince yourself that $\rho(A_{1},\dots,A_{r})\geq\rho(A_{1}+\dots+A_{r})$. When is $\rho(A_{1},\dots,A_{r})=\rho(A_{1}+\dots+A_{r})$? I am also interested for characterizations of when $\rho(A_{1},\dots,A_{r})=\rho(A_{1}+\dots+A_{r})$ when $A_{1},\dots,A_{r}$ satisfy some conditions such as normality.
