On the essential spectra of Toeplitz operators on Bergman spaces of the bi-disc

In this note we deal with the problem of determining the essential spectrum of a Toeplitz operator $T_f:A^2({\mathbb{D}}^2)\to A^2({\mathbb{D}}^2)$ acting on the Bergman space $A^2({\mathbb{D}}^2)$ of the bi-disc whose symbol $f\in C(\overline{{\mathbb{D}}^2})$ lies in the space of continuous functions on the closure $\overline{{\mathbb{D}}^2}$ of the bi-disc.