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The Bargmann transform is a transform which maps Fourier-invariant function spaces and their duals to certain spaces of formal power series expansions, which sometimes are convenient classes of analytic functions.

In the 70th, Berezin used the Bargmann transform to translate problems in operator theory into an analytic pseudo-differential calculus, the so-called Wick calculus, where the involved symbols are analytic functions, and the corresponding operators map suitable classes of entire functions into other classes of entire functions. In the same manner, the Toeplitz operators correspond to so-called anti-Wick operators on the Bargmann transformed side.

Recently, some investigations on certain Fourier invariant subspaces of the Schwartz space and their dual (distribution) spaces have been performed by the author. These spaces are called Pilipovi ́c spaces, and are defined by imposing suitable boundaries on the Hermite coefficients of the involved functions or distributions. The family of Pilipovi ́c spaces contains all Fourier invariant Gelfand-Shilov spaces as well as other spaces which are strictly smaller than any Fourier invariant non-trivialGelfand-Shilov space. In the same way, the family of Pilipovi ́c distribution spaces contains spaces which are strictly larger than any Fourier invariant Gelfand-Shilov distribution space.

In the talk we show that the Bargmann images of Pilipovi ́c spaces and their distribution spaces are convenient classes of analytic functions or power series expansions which are suitable when investigating Wick operators (i. e. the operators in the Wick calculus).

We deduce continuity properties for such operators when the symbols and target functions possess certain (weighted) Lebesgue estimates. We also explain how the counter images with respect to the Bargmann transform of these results generalise some continuity results for (real) pseudo-differential operators with symbols in modulation spaces, when acting on other modulation space. Finally we discuss some links between ellipticity in the real pseudo-differential calculus and the Wick calculus, as well as links between Wick and anti-Wick operators.

The talk is based on collaborations with Nenad Teofanov and Patrik Wahlberg, and parts of the content of the talk is available at:

N. Teofanov, J. Toft Pseudo-differential calculus in a Bargmann setting, Ann. Acad. Sci. Fenn.

Math. 45 (2020), 227–257.

N. Teofanov, J. Toft, P. Wahlberg Pseudo-differential operators with isotropic symbols, and Wick and anti-Wick operators, arXiv:??