Using the Schur–Weyl duality and Fock-type tensor calculus, it is proved that each function f from Gaussian Hilbert complex space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1674_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with Haar’s probability measure on the unitary group U(N) has unique expansion <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1674_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {S}f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> belonging to the Hardy–Dwyer space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1674_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> of entire Hilbert-Schmidt holomorphic functions on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1674_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </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>. The transform <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1674_Article_IEq5.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {S}:{L}^2\longrightarrow \mathcal {H}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">S</mi> <mo>:</mo> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">⟶</mo> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> provides a unitary surjective isomorphism. Applications are concerned with non-commutative operator Weyl pairs in quantum theory.

Norm-Preserving Holomorphic Expansions of Random Functions on Unitary Matrix Groups

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发表日期：

2025年3月14日

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Oleh Lopushansky

摘要

Using the Schur–Weyl duality and Fock-type tensor calculus, it is proved that each function f from Gaussian Hilbert complex space $L^2$ $L^{2}$ with Haar’s probability measure on the unitary group U(N) has unique expansion $\mathcal {S}f$ $S f$ belonging to the Hardy–Dwyer space $\mathcal {H}^2$ $H^{2}$ of entire Hilbert-Schmidt holomorphic functions on $\mathbb {C}^N$ $C^{N}$ . The transform $\mathcal {S}:{L}^2\longrightarrow \mathcal {H}^2$ $S : L^{2} ⟶ H^{2}$ provides a unitary surjective isomorphism. Applications are concerned with non-commutative operator Weyl pairs in quantum theory.