We give a fundament for Berezin's analytic Psi do considered in [4] in terms of Bargmann images of Pilipovic spaces. We deduce basic continuity results for such Psi do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply wellknown continuity results for real Psi do with symbols in modulation spaces, when acting on other modulation spaces.