A Wiener algebra for Fock space operators

We introduce an algebra ${\mathcal{W}}_t$ of linear operators that act continuously on each of the Fock spaces $F_t^p$, $1\le p\le \mathrm{\infty }$, 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 $p$.