Root test for matrix series via spectral radius?

Let $\sum_{k=1}^\infty A_k$ be a matrix series in $\mathrm{M}_n(\mathbb{C})$ and let $||\cdot||$ be a submultiplicative norm on $\mathrm{M}_n(\mathbb{C})$. Since the root test is valid in any Banach space $(E,||\cdot||)$, we can consider the quantity $$a:=\limsup_{n\to\infty}||A_n||^{1/n}$$ (which btw does not depend on the choice of norm by topological equivalence of all norms on the finite-dimensional vector space $\mathrm{M}_n(\mathbb{C})$). Let us denote by $\rho$ the spectral radius (map) on $\mathrm{M}_n(\mathbb{C})$. Since we have chosen $||\cdot||$ to be submultiplicative on $\mathrm{M}_n(\mathbb{C})$, we have $\forall A\in\mathrm{M}_n(\mathbb{C}): \rho(A)\leq ||A||$, hence $$\limsup_{n\to\infty}\rho(A_n)^{1/n}\leq a$$

Is it true that $a=\limsup\limits_{n\to\infty}\rho(A_n)^{1/n}$ ? If not, what would be some counter-example(s)?

Apologies, if this is too elementary. I am a novice to matrix analysis and some (admittedly very) simple examples suggest this might be true (though I am afraid the choice of examples could just be an expression of my own confirmation bias). Thanks!

• The condition will hold if we have $A_k = A^k$ for some fixed matrix $A$; I'm not sure what else you might have in mind Commented Jul 2, 2018 at 5:08

It is not generally true that $a = \limsup_{n \to \infty} \rho(A_n)^{1/n}$. As a simple example, consider the series given by $$A_k = \pmatrix{0&2^{-k}\\0&0}$$ We find that $a = \frac 12$, but $\rho(A_n) = 0$ for all $n$.