Quantum Harmonic Analysis on the Unweighted Bergman Space of the Unit Ball

TL;DR

This paper explores quantum harmonic analysis on the Bergman space of the unit ball in complex n-dimensional space, establishing key theorems and characterizations related to Toeplitz operators and the Berezin transform.

Contribution

It introduces a Wiener's Tauberian theorem and characterizes the radial Toeplitz algebra on the Bergman space, advancing understanding of operator approximations.

Findings

Formulated a Wiener's Tauberian theorem for the setting.

Provided characterizations of the radial Toeplitz algebra.

Investigated approximation properties via Toeplitz operators.

Abstract

We study quantum harmonic analysis (QHA) on the Bergman space $\mathcal{A}^2(\mathbb{B}^n)$ over the unit ball in $\mathbb{C}^n$. We formulate a Wiener's Tauberian theorem, and characterizations of the radial Toeplitz algebra over $\mathcal{A}^2(\mathbb{B}^n)$. We discuss the $\alpha$-Berezin transform and investigate the question of approximations by Toeplitz operators.