Pseudodifferential operators with completely periodic symbols

Motivated by the recent paper of Boggiatto-Garello in J. Pseudo-Differ. Oper. Appl. $\mathbf{\text{11}}$ (2020), 93-117, where a Gabor operator is regarded as pseudodifferential operator with symbol $p(x,\omega )$ periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general $\tau$ quantization.