Quantum Harmonic Analysis on the Unweighted Bergman Space of the Unit Ball
Matthew Dawson, Vishwa Dewage, Mishko Mitkovski, Gestur Olafsson
TL;DR
The paper develops a quantum harmonic analysis framework for the Bergman space $\mathcal{A}^2(\mathbb{B}^n)$ by leveraging scalar-type holomorphic discrete series representations of $\mathrm{SU}(n,1)$. It provides a unified treatment of left translations, convolutions, and radial structures to characterize Toeplitz algebras, formulate a Wiener's Tauberian theorem for operator and Schatten-class settings, and introduce an $\alpha$-Berezin transform that connects Toeplitz operators with their spectral data through radial convolutions. The main contributions include: (i) a Tauberian criterion for cyclic vectors in the QHA context, (ii) a description of the radial Toeplitz algebra as left-uniformly continuous radial operators, (iii) a natural $\alpha$-Berezin transform $\widetilde{B}_\alpha$ that yields convergence $T_{\widetilde{B}_\alpha(S)}\to S$ in operator norm for a broad class of $S$, and (iv) a detailed analysis of radial convolutions and their impact on the Toeplitz calculus on the unit ball. These results advance a systematic noncommutative harmonic-analysis approach to Toeplitz operators on Bergman spaces with potential extensions to broader domains and representations.
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 $α$-Berezin transform and investigate the question of approximations by Toeplitz operators.