  We introduce an algebra $\mathcal W_t$ of linear operators that act
continuously on each of the Fock spaces $F_t^p$, $1 \leq p \leq \infty$, and
contains all Toeplitz operators with bounded symbols. We show that compactness,
the spectrum, essential spectrum and the Fredholm index of an element of
$\mathcal W_t$, realized as an operator on $F_t^p$, are independent of the
value of $p$.
