The purpose of this note is to characterize the reducibility and the n-hypercontractivity of extensions of Cowen–Douglas operators, and to show that the two have a mutually determining relationship in this context. In doing so, the curvature of the Hermitian holomorphic vector bundle is considered. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_421_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}B_k(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msub> <mi>B</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the class of Cowen–Douglas operators with flag structure and index k. As an important class of geometric operators, these operators have been studied extensively in recent research. It has been proven to be norm dense in the Cowen–Douglas operator class with index k. As applications, we provide a sufficient condition that operators in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_421_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}B_k(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msub> <mi>B</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are not n-hypercontractive.

A note on reducibility and n-hypercontractivity of extensions of Cowen–Douglas operators

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发表日期：

2025年5月5日

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Shanshan Ji

摘要

The purpose of this note is to characterize the reducibility and the n-hypercontractivity of extensions of Cowen–Douglas operators, and to show that the two have a mutually determining relationship in this context. In doing so, the curvature of the Hermitian holomorphic vector bundle is considered. Let $\mathcal {F}B_k(\Omega )$ $F B_{k} (Ω)$ denote the class of Cowen–Douglas operators with flag structure and index k. As an important class of geometric operators, these operators have been studied extensively in recent research. It has been proven to be norm dense in the Cowen–Douglas operator class with index k. As applications, we provide a sufficient condition that operators in $\mathcal {F}B_k(\Omega )$ $F B_{k} (Ω)$ are not n-hypercontractive.