Assume that $T:\ell^1 \rightarrow \ell^1$ is a bounded linear operator, given by an infinite matrix $(a_{ij})_{i,j\in\mathbb{N}}$ so that $(Tx)_i = \sum_{j=1}^{\infty} a_{ij}x_j$. In particular, I am interested in nonnegative matrices ($a_{ij}\geq0$) such that $a_{ii} = 0$ and $a_{ij}>0$ only for $i-m \leq j \leq i+m$, $j \neq i$, where $m$ is a constant.

I would like to understand when such $T$ is quasi-compact. Are there sufficient or necessary conditions in terms of the matrix entries? What would be a good intuition? How should a counter-example look like?