Quantum harmonic analysis for polyanalytic Fock spaces

TL;DR

This paper extends quantum harmonic analysis to polyanalytic Fock spaces, providing new insights into Toeplitz operators, their kernels, and the Berezin transform, with implications for operator theory in these spaces.

Contribution

It introduces a quantum harmonic analysis framework for reducible settings, characterizes Toeplitz operator kernels, and explores the properties of the Berezin transform on polyanalytic Fock spaces.

Findings

Existence of symbols with unitary Toeplitz operators on analytic but vanishing on polyanalytic spaces

Explicit kernel characterization of Toeplitz quantization

Injectivity of the Berezin transform on Toeplitz operators

Abstract

We develop the quantum harmonic analysis framework in the reducible setting and apply our findings to polyanalytic Fock spaces. In particular, we explain some phenomena observed in arXiv:2201.10230 and answer a few related open questions. For instance, we show that there exists a symbol such that the corresponding Toeplitz operator is unitary on the analytic Fock space but vanishes completely on one of the true polyanalytic Fock spaces. This follows directly from an explicit characterization of the kernel of the Toeplitz quantization, which we derive using quantum harmonic analysis. Moreover, we show that the Berezin transform is injective on the set of of Toeplitz operators. Finally, we provide several characterizations of the $\mathcal{C}_1$-algebra in terms of integral kernel estimates and essential commutants.