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<p>By applying the distribution approach and the Duhamel products method, we calculate the spectral multiplicity of direct sums of operators of the forms <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10154_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( S\oplus \left( J^{\alpha }+I\right) \right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>S</mi> <mo>⊕</mo> <mfenced close=")" open="("> <msup> <mi>J</mi> <mi>α</mi> </msup> <mo>+</mo> <mi>I</mi> </mfenced> </mfenced> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10154_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( W_{zw}\oplus A\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>W</mi> <mrow> <mi mathvariant="italic">zw</mi> </mrow> </msub> <mo>⊕</mo> <mi>A</mi> </mfenced> </math></EquationSource> </InlineEquation>, where <i>S</i> is the shift operator, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10154_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(J^{\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>J</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> is the Riemann-Liouville fractional integration operator and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10154_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(W_{zw}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>W</mi> <mrow> <mi mathvariant="italic">zw</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the restricted double integration operator.</p>

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Some Multiplicity Results for Integration and Fractional Integration Operators

  • 期刊论文

  • 发表日期：

    2025年4月17日

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  • Mubariz T. Garayev

摘要

By applying the distribution approach and the Duhamel products method, we calculate the spectral multiplicity of direct sums of operators of the forms \(\left( S\oplus \left( J^{\alpha }+I\right) \right) \) S ⊕ J α + I and \(\left( W_{zw}\oplus A\right) \) W zw ⊕ A , where S is the shift operator, \(J^{\alpha }\) J α is the Riemann-Liouville fractional integration operator and \(W_{zw}\) W zw is the restricted double integration operator.