In this paper, we prove that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R=T_1+T_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is subscalar of order 8, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_1,T_2\in \mathcal {L}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="script">L</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are complex symmetric or J-self-adjoint operators with property <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T_1T_2=\lambda T_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>λ</mi> <msub> <mi>T</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda \in \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation>. In particular, when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we further show that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R+A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>+</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> is decomposable for every algebraic operator <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A\in \mathcal {L}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> commuting with R. As an application, we verify that if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$T\in \mathcal {L}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a C-symmetric operator with property <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(\delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and either u or Cv is an eigenvector of T, then <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$T+u{\otimes }v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>+</mo> <mi>u</mi> <mo>⊗</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation> is decomposable if and only if T is decomposable. We also study the decomposability of operator matrices.

Local spectral properties for sums of operators related to complex symmetry

期刊论文
发表日期：

2026年7月6日

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Sungeun Jung

摘要

In this paper, we prove that $R=T_1+T_2$ $R = T_{1} + T_{2}$ is subscalar of order 8, where $T_1,T_2\in \mathcal {L}(\mathcal {H})$ $T_{1}, T_{2} \in L (H)$ are complex symmetric or J-self-adjoint operators with property $(\delta )$ $(δ)$ and $T_1T_2=\lambda T_1$ $T_{1} T_{2} = λ T_{1}$ for some $\lambda \in \mathbb {C}$ $λ \in C$ . In particular, when $\lambda =0$ $λ = 0$ , we further show that $R+A$ $R + A$ is decomposable for every algebraic operator $A\in \mathcal {L}(\mathcal {H})$ $A \in L (H)$ commuting with R. As an application, we verify that if $T\in \mathcal {L}(\mathcal {H})$ $T \in L (H)$ is a C-symmetric operator with property $(\delta )$ $(δ)$ and either u or Cv is an eigenvector of T, then $T+u{\otimes }v$ $T + u \otimes v$ is decomposable if and only if T is decomposable. We also study the decomposability of operator matrices.