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<p>For <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(M_C=\left( \begin{array}{cccc}A&amp; C\\ 0&amp; B\end{array}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>C</mi> </msub> <mo>=</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mn>0</mn> </mrow> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> acting on a Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({\mathcal{H}}\oplus {\mathcal{K}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo>⊕</mo> <mi mathvariant="script">K</mi> </mrow> </math></EquationSource> </InlineEquation>, we first characterize the Fredholm completions with positive nullity and negative index. We then explore the weak approximate spectrum <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\sigma _{_\textrm{Fa}}(M_C)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>σ</mi> <mmultiscripts> <mrow /> <mtext>Fa</mtext> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>C</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the weak essential approximate spectrum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\sigma _{_\textrm{Fea}}(M_C)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>σ</mi> <mmultiscripts> <mrow /> <mtext>Fea</mtext> <mrow /> </mmultiscripts> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>C</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(M_C\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>C</mi> </msub> </math></EquationSource> </InlineEquation>. In combination with the research, we give the equivalent conditions that make <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(M_C\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>C</mi> </msub> </math></EquationSource> </InlineEquation> have the weak property <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\((\omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(C\in {\mathcal{B}}({\mathcal{K}},{\mathcal{H}}).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">K</mi> <mo>,</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Weak property \((\omega )\) for two-by-two operator matrices

  • 期刊论文

  • 发表日期：

    2025年11月16日

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  • Jiong Dong

摘要

For \(M_C=\left( \begin{array}{cccc}A& C\\ 0& B\end{array}\right)\) M C = A C 0 B acting on a Hilbert space \({\mathcal{H}}\oplus {\mathcal{K}}\) H ⊕ K , we first characterize the Fredholm completions with positive nullity and negative index. We then explore the weak approximate spectrum \(\sigma _{_\textrm{Fa}}(M_C)\) σ Fa ( M C ) and the weak essential approximate spectrum \(\sigma _{_\textrm{Fea}}(M_C)\) σ Fea ( M C ) of \(M_C\) M C . In combination with the research, we give the equivalent conditions that make \(M_C\) M C have the weak property \((\omega )\) ( ω ) for any \(C\in {\mathcal{B}}({\mathcal{K}},{\mathcal{H}}).\) C ∈ B ( K , H ) .