The main goal of this paper is to study some local spectral properties of the generalized derivation operator and the elementary multiplication operator. More precisely, let A, B be bounded operators on a Banach space E, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="226" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _{A, B}: X\in \mathcal {L}(E)\rightarrow AX-XB$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>A</mi> <mi>X</mi> <mo>-</mo> <mi>X</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="195" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{A, B}: X \in \mathcal {L}(E) \rightarrow AXB$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>A</mi> <mi>X</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> be the generalized derivation and the elementary multiplication operator, respectively. We investigate the relation between point spectrum of A, B and those of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _{A, B}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{A, B}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We also consider the set where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta _{A, B}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{A, B}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> fail to have (SVEP), and Bishop property <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2962_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Some recent results from the literature are also recaptured.

Some Local Spectral Properties of Elementary Operators $\delta _{A, B}$ and $\Delta _{A, B}$

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发表日期：

2025年10月3日

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Chaimaa Benzarouala,
Nadia Ourchane,
El Hassan Zerouali

摘要

The main goal of this paper is to study some local spectral properties of the generalized derivation operator and the elementary multiplication operator. More precisely, let A, B be bounded operators on a Banach space E, $\delta _{A, B}: X\in \mathcal {L}(E)\rightarrow AX-XB$ $δ_{A, B} : X \in L (E) \to A X - X B$ , and $\Delta _{A, B}: X \in \mathcal {L}(E) \rightarrow AXB$ $Δ_{A, B} : X \in L (E) \to A X B$ be the generalized derivation and the elementary multiplication operator, respectively. We investigate the relation between point spectrum of A, B and those of $\delta _{A, B}$ $δ_{A, B}$ and $\Delta _{A, B}$ $Δ_{A, B}$ . We also consider the set where $\delta _{A, B}$ $δ_{A, B}$ and $\Delta _{A, B}$ $Δ_{A, B}$ fail to have (SVEP), and Bishop property $(\beta )$ $(β)$ . Some recent results from the literature are also recaptured.

Some Local Spectral Properties of Elementary Operators \(\delta _{A, B}\) and \(\Delta _{A, B}\)

摘要