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<p>Let <i>T</i> be a bounded right linear operator acting on a right quaternionic Hilbert space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1676_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. We present several properties of the analytic core <i>K</i>(<i>T</i>) and the quasi-nilpotent part <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1676_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\( {\mathcal {H}}_{0}(T)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <i>T</i>. Furthermore, for a quaternion <i>q</i>, we investigate the case where the equivalence class [<i>q</i>] is an isolated part in the S-spectrum of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1676_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\( T \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>T</mi> </math></EquationSource> </InlineEquation>.</p>

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On Isolated Parts of the S-Spectrum of a Bounded Right Linear Operator

  • 期刊论文

  • 发表日期：

    2025年3月11日

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  • Rachid Arzini,
  • Ali Jaatit

摘要

Let T be a bounded right linear operator acting on a right quaternionic Hilbert space \({\mathcal {H}}\) H . We present several properties of the analytic core K(T) and the quasi-nilpotent part \( {\mathcal {H}}_{0}(T)\) H 0 ( T ) of T. Furthermore, for a quaternion q, we investigate the case where the equivalence class [q] is an isolated part in the S-spectrum of \( T \) T .