We introduce a Banach algebra \(\mathcal W_t\) of integral kernels over \(\mathbb C^n\) . An integral operators with such a kernel acts continuously on each of the Fock spaces \(F_t^p\) , \(1 \leq p \leq \infty \) . The class of these operators 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.