Pseudodifferential operators with completely periodic symbols

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