# Berger-Coburn theorem, localized operators, and the Toeplitz algebra

## TL;DR

This paper simplifies the proof of the Berger-Coburn theorem on Toeplitz operator boundedness, extends it to p-Fock spaces, and introduces new characterizations of the Toeplitz $C^*$ algebra based on recent compactness results.

## Contribution

It provides a simplified proof of a classical theorem, extends the theorem to p-Fock spaces, and offers three new characterizations of the Toeplitz $C^*$ algebra.

## Key findings

- Extended Berger-Coburn theorem to p-Fock spaces.
- Presented three new characterizations of the Toeplitz $C^*$ algebra.
- Reviewed recent results on compactness characterization via Berezin transform.

## Abstract

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend this theorem to the setting of $p$-Fock spaces $(1\leq p \leq \infty)$. We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz $C^*$ algebra generated by Toeplitz operators with bounded symbols.