Toeplitz Operators on Fock Space $F_{\alpha}^{\infty}$

TL;DR

This paper characterizes when measures induce bounded and compact Toeplitz operators on Fock space, using Berezin transforms, and explores symbols in BMO for such operators.

Contribution

It provides necessary and sufficient conditions for Fock-Carleson measures and characterizes Toeplitz operators with BMO symbols on Fock space.

Findings

Criteria for boundedness of Toeplitz operators

Conditions for compactness of Toeplitz operators

Characterization of BMO symbols in this context

Abstract

In this paper, we study necessary and sufficient conditions for a positive Borel measure $\mu$ on the complex space $\mathbb{C}$ to be a $(\infty,q)$ or $(p,\infty)$ (vanishing) Fock-Carleson measure through its Berezin transform. Then we discuss boundedness and compactness of the Toeplitz operator $T_{\mu}$ with a positive Borel measure $\mu$ as symbol on Fock space $F_{\alpha}^{\infty}$. Furthermore, we charaterize these properties of the Toeplitz operator T_{\varphi}$ with a symbol $\varphi$ which is in $BMO$.