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arXiv · 2610.33428

Quantum harmonic analysis and the Differential operators on weighted Bergman spaces of bounded symmetric domains

Matthew Dawson·Vishwa Dewage·Gestur Olafsson
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Abstract

We develop quantum harmonic analysis (QHA) for operators on the parametrized Bergman space $\mathcal{A}^2_ν(Ω)$ of a bounded symmetric domain $Ω$, extending the theory previously known for the unit ball. Building on this framework, we introduce a class of differential operators acting on operators on $\mathcal{A}^2_ν(Ω)$. Combining these tools, we advance the theory of Toeplitz operators on $\mathcal{A}^2_ν(Ω)$: we establish Toeplitz approximation results via the $α$-Berezin transform, construct a differential operator $D_α$ that shifts the $α$-Berezin transform, and obtain several characterizations of the radial Toeplitz algebra -- including one via the uniformly continuous radial operators, and one via the domain of $D_α$.

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Matthew Dawson, Vishwa Dewage, Gestur Olafsson. 2026-10-24. Quantum harmonic analysis and the Differential operators on weighted Bergman spaces of bounded symmetric domains. https://arxiv.org/abs/2610.33428

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