Divergence of power series in commutative Banach algebras

Let $(A,||\cdot||)$ be a commutative Banach algebra over $\mathbb{C}$. Consider a formal power series $f(z):=\sum_{n=0}a_n z^n\in A[[z]]$ and let $$r:=\frac{1}{\limsup\limits_{n\to\infty}||a_n||^{1/n}}$$ For $b\in A$ denote by $\rho(b)$ its spectral radius.

Is it generally true that $f(b)$ is divergent (in the $||\cdot||$-topology) for any $b\in A$ with $\rho(b)>r$?

I believe one can easily show in certain cases that $f(b)$ is divergent if $$\rho(b)> \frac{1}{\limsup\limits_{n\to\infty}\rho(a_n)^{1/n}}=:R,$$ but $R$ can be much bigger than $r$.

No, and more generally you can make no statement along these lines that depends only on $\rho(b)$. For instance, let $a,b\in A$ be elements with nonzero spectral radius such that $ab=0$. Let $a_n=c_na$ for some scalars $c_n$. Then $r$ can be made arbitrary by choosing $c_n$ appropriately, but $f(b)$ will always converge since all terms after the first term will be $0$.
• I see. The examples I had in mind, where the $\rho(b)$-thing works, are indeed too special. A follow-up question if you don't mind (let me know if it is better to create a new topic for it): based on $r$ is it at all possible to conclude where $f$ diverges, or that works only in special cases? (I mean aside from simply restating the root criterion.)
• I'm not sure exactly what you mean by that. The problem with getting any sort of divergence statement is that there is no lower bound for $\|a_nb^n\|$ in terms of just numerical data about $a_n$ and $b$ individually (you also need to know something about their product, which could be smaller than expected). Jul 2, 2018 at 23:31
• Yes, this was what I meant. In $\mathbb{C}$ everything works out fine because the absolute value is multiplicative, but in general $A$ the norm is only submultiplicative, and for example, if I apply the root test, I can only conclude convergence, but the norm inequality is in the "wrong" direction in order to conclude divergence. Do you happen to have any references where such topics and examples about convergence/divergence issues of power series with Banach algebra coefficients are handled? I would really like to learn more about this topic, but I don't really know the literature.