  We analyse the $\ell^2(\pi)$-convergence rate of irreducible and aperiodic
Markov chains with $N$-band transition probability matrix $P$ and with
invariant distribution $\pi$. This analysis is heavily based on: first the
study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(\pi)})$ of
$P\_{|\ell^2(\pi)}$ derived from Hennion's quasi-compactness criteria; second
the connection between the spectral gap property (SG$\_2$) of $P$ on
$\ell^2(\pi)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$)
is shown to hold under the condition \[\alpha\_0 := \sum\_{{m}=-N}^N
\limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\,
1. \] Moreover $r\_{ess}(P\_{|\ell^2(\pi)}) \leq \alpha\_0$. Simple conditions
on asymptotic properties of $P$ and of its invariant probability distribution
$\pi$ to ensure that $\alpha\_0\textless{}1$ are given. In particular this
allows us to obtain estimates of the $\ell^2(\pi)$-geometric convergence rate
of random walks with bounded increments. The specific case of reversible $P$ is
also addressed. Numerical bounds on the convergence rate can be provided via a
truncation procedure. This is illustrated on the Metropolis-Hastings algorithm.
