<p>We study quantum harmonic analysis (QHA) on the Bergman space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2803_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}^2(\mathbb {B}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over the unit ball in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2803_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We formulate a Wiener’s Tauberian theorem, and characterizations of the radial Toeplitz algebra over <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2803_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}^2(\mathbb {B}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">A</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We discuss the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2803_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Berezin transform and investigate the question of approximations by Toeplitz operators.</p>

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

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2025年7月19日

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Matthew Dawson,
Vishwa Dewage,
Mishko Mitkovski,
Gestur Ólafsson

摘要

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