Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

We study the link between Ψdos and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of Ψdos of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in terms of anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.