Scholarlink

<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {G}_1, \mathcal {G}_2\)</EquationSource> </InlineEquation> be subsets of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {L}(\textbf{X})\)</EquationSource> </InlineEquation> that contain all rank-one idempotent and nilpotent operators. For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon &gt; 0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textbf{T} \in \mathcal {L}(\textbf{X})\)</EquationSource> </InlineEquation>, we denote by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\sigma _{\varepsilon }(\textbf{T})\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(r_{\varepsilon }(\textbf{T})\)</EquationSource> </InlineEquation> <i>the pseudospectrum</i> and <i>the pseudospectral radius</i> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textbf{T}\)</EquationSource> </InlineEquation>, respectively. In this work, we characterize the structure of surjective mappings <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\varphi : \mathcal {G}_1 \rightarrow \mathcal {G}_2\)</EquationSource> </InlineEquation> satisfying <Equation ID="Equa"> <EquationSource Format="TEX">\( r_\varepsilon (\varphi (\textbf{T}_1)\varphi (\textbf{T}_2)\varphi (\textbf{T}_1)) = r_\varepsilon (\textbf{T}_1 \textbf{T}_2 \textbf{T}_1) \quad {\text {or}} \quad \sigma _\varepsilon (\varphi (\textbf{T}_1)\varphi (\textbf{T}_2)\varphi (\textbf{T}_1)) = \sigma _\varepsilon (\textbf{T}_1 \textbf{T}_2 \textbf{T}_1), \)</EquationSource> </Equation>for all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\textbf{T}_1, \textbf{T}_2 \in \mathcal {G}_1\)</EquationSource> </InlineEquation>. Analogous descriptions are also established in the finite-dimensional setting without requiring the surjectivity of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> </InlineEquation>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Non-linear maps preserving the pseudo spectral radius of Jordan triple products on Banach spaces

  • 期刊论文

  • 发表日期：

    2025年12月3日

查看全文
  • Anas Cade,
  • Azzedine EL Asri

摘要

Let \(\mathcal {G}_1, \mathcal {G}_2\) be subsets of \(\mathcal {L}(\textbf{X})\) that contain all rank-one idempotent and nilpotent operators. For \(\varepsilon > 0\) and \(\textbf{T} \in \mathcal {L}(\textbf{X})\) , we denote by \(\sigma _{\varepsilon }(\textbf{T})\) and \(r_{\varepsilon }(\textbf{T})\) the pseudospectrum and the pseudospectral radius of \(\textbf{T}\) , respectively. In this work, we characterize the structure of surjective mappings \(\varphi : \mathcal {G}_1 \rightarrow \mathcal {G}_2\) satisfying \( r_\varepsilon (\varphi (\textbf{T}_1)\varphi (\textbf{T}_2)\varphi (\textbf{T}_1)) = r_\varepsilon (\textbf{T}_1 \textbf{T}_2 \textbf{T}_1) \quad {\text {or}} \quad \sigma _\varepsilon (\varphi (\textbf{T}_1)\varphi (\textbf{T}_2)\varphi (\textbf{T}_1)) = \sigma _\varepsilon (\textbf{T}_1 \textbf{T}_2 \textbf{T}_1), \) for all \(\textbf{T}_1, \textbf{T}_2 \in \mathcal {G}_1\) . Analogous descriptions are also established in the finite-dimensional setting without requiring the surjectivity of \(\varphi \) .