<p>We classify self-adjoint first-order differential operators on weighted Bergman spaces on the <i>N</i>-dimensional unit ball <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {B}}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {D}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> complex matrices satisfying <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$I-Z^*Z>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <msup> <mi>Z</mi> <mo>∗</mo> </msup> <mi>Z</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Our main tools are the discrete series representations of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SU}(N,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_160_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SU}(2,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>SU</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We believe that our observations will extend to general bounded symmetric domains.</p>

Self-adjoint differential operators on Bergman space on bounded symmetric domains

期刊论文
发表日期：

2025年7月4日

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Jens Gerlach Christensen,
Christopher Benjamin Deng

摘要

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the N-dimensional unit ball ${\mathbb {B}}^N$ $B^{N}$ and ${\mathbb {D}}^2$ $D^{2}$ of $2\times 2$ $2 \times 2$ complex matrices satisfying $I-Z^*Z>0$ $I - Z^{*} Z > 0$ . Our main tools are the discrete series representations of $\textrm{SU}(N,1)$ $SU (N, 1)$ and $\textrm{SU}(2,2)$ $SU (2, 2)$ . We believe that our observations will extend to general bounded symmetric domains.