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<p>In this paper, we study basic properties of compact operators on Hilbert C*-modules over an arbitrary finite dimensional C*-algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_461_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. We introduce the notions of invariant and hyperinvariant submodules in the setting of Hilbert C*-modules and prove a Lomonosov type theorem for compact modular operators on such modules. Specifically, we show that every nonzero compact modular operator acting on a Hilbert <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_461_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-module admits a proper nonzero hyperinvariant submodule.</p>

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Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules

  • 期刊论文

  • 发表日期：

    2025年7月3日

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  • Kamran Sharifi

摘要

In this paper, we study basic properties of compact operators on Hilbert C*-modules over an arbitrary finite dimensional C*-algebra \(\mathcal {A}\) A . We introduce the notions of invariant and hyperinvariant submodules in the setting of Hilbert C*-modules and prove a Lomonosov type theorem for compact modular operators on such modules. Specifically, we show that every nonzero compact modular operator acting on a Hilbert \(\mathcal {A}\) A -module admits a proper nonzero hyperinvariant submodule.