  In this paper we study the Fredholm properties of Toeplitz operators acting
on weighted Bergman spaces $A^p_{\nu}(\mathbb{B}^n)$, where $p \in (1,\infty)$
and $\mathbb{B}^n \subset \mathbb{C}^n$ denotes the $n$-dimensional open unit
ball. Let $f$ be a continuous function on the Euclidean closure of
$\mathbb{B}^n$. It is well-known that then the corresponding Toeplitz operator
$T_f$ is Fredholm if and only if $f$ has no zeros on the boundary
$\partial\mathbb{B}^n$. As a consequence, the essential spectrum of $T_f$ is
given by the boundary values of $f$. We extend this result to all operators in
the algebra generated by Toeplitz operators with bounded symbol (in a sense to
be made precise down below). The main ideas are based on the work of Suarez et
al. and limit operator techniques coming from similar problems on the sequence
space $\ell^p(\mathbb{Z})$.
