Let $k, a, b > 0$ and $b < 1$. Define the $n \times n$ Hessenberg Toeplitz matrix
$$ {\bf A}_n := \begin{bmatrix} ab & ab^2 & ab^3 & \cdots & ab^n \\ k & ab & ab^2 & \cdots & ab^{n-1} \\ 0 & k & ab & \cdots & ab^{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & ab \end{bmatrix} $$
I would like to know how one can calculate the spectral radius of the infinite matrix one obtains after letting $n \to \infty$. I am interested when the limit is in $\ell^{2}(\mathbb N)$, i.e., I am interested only in one-sided infinite matrices.
I know even in this case the symbol of the limiting matrix will give me the operator norm for the corresponding infinite Toeplitz matrix, which merely gives an upper bound for the spectral radius. I've checked Arveson's Short course on spectral theory and others, but was not able to find an answer. Any help would be appreciated!
Motivation
I came across this matrix while working on some discrete dynamical system. It has to do with the contraction of the system. Basically, the spectral radius of the infinite Toeplitz matrix ${\bf A}_\infty$ will correspond to the contraction rate of the system (if $\rho ({\bf A}_\infty) < 1$).
