On Toeplitz operators between Fock spaces

TL;DR

This paper characterizes the boundedness and compactness of Toeplitz operators between Fock spaces using Berezin transforms and averaging functions, providing asymptotic norm estimates and extending recent research.

Contribution

It offers a comprehensive description of Toeplitz operators on Fock spaces in terms of measures and transforms, filling gaps in existing literature.

Findings

Characterization of bounded Toeplitz operators via Berezin transforms

Asymptotic estimates for operator norms

Extension of previous results to all Banach-Fock spaces

Abstract

We study mapping properties of Toeplitz operators $T_\mu$ associated to nonnegative Borel measure $\mu$ on the complex space $\mathbb{C}^n$. We, in particular, describe the bounded and compact operators $T_\mu$ acting between Fock spaces in terms of the objects $t$-Berezin transforms, averaging functions, and averaging sequences of their inducing measures $\mu$. An asymptotic estimate for the norms of the operators has been also obtained. The results obtained extend a recent work of Z. Hu and X. Lv and fills the remaining gap when both the smallest and largest Banach--Fock spaces are taken into account.