有限维向量空间中无特征值的算子：线性化与谱等价
Operators without eigenvalues in finite-dimensional vector spaces: Linearization and Spectral Equivalence
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中文总结 AI 辅助
研究有限维向量空间中一类闭对称线性关系\(S\)的自伴扩张，通过引入矩阵多项式类\(\mathbb{P}_{\mathsf Q}\)，建立其与\(S\)自伴扩张类\(\mathbb A_{S,\mathsf{b}}\)等价类的双射，还给出线性化与边界特征值问题谱等价的条件。
中文摘要 AI 辅助
本文中，\(S\)是克莱因空间\(\mathfrak H\)中的一个闭对称线性关系，其伴随为\(S^*\)，亏数有限且相等为\(d\)，还有一个边界映射\(\mathsf{b}: S^*\rightarrow \mathbb{C}^{2d}\)，其格拉姆矩阵为\(\mathsf Q\)。我们引入了\(S\)在包含\(\mathfrak H\)作为有限余维克莱子空间的克莱因空间\(\widetilde{\mathfrak H}\)中的自伴扩张类\(\mathbb A_{S,\mathsf{b}}\)，以及\(d\times 2d\)矩阵多项式类\(\mathbb{P}_{\mathsf Q}\)。这两类都配备了自然等价关系。给定\(\mathcal{P}(z) \in \mathbb{P}_{\mathsf Q}\)，我们考虑由条件\(\mathcal{P}(z)\mathsf{b}(\{f,g\}) = 0\)，\(\{f,g\}\in S^*\)定义的边界特征值问题，并寻找其线性化。线性化是指线性关系\(\widetilde{A} \in \mathbb{A}_{S, \mathsf b}\)，使得由\(\widetilde A\)确定的\(S\)的施特劳斯扩张\(T_{\widetilde A}(z)\)对所有\(z \in \overline{\mathbb C}\)都与线性关系\(\big\{\{f,g\}\in S^*\,:\,\mathcal P(z)\mathsf b(\{f,g\})=0\big\}\)一致。我们证明这种对应在\(\mathbb A_{S, \mathsf b}\)和\(\mathbb P_{\mathsf Q}\)的等价类之间定义了一个双射。此外，我们提供了一个条件，在此条件下所得线性化与边界特征值问题谱等价：它们的正则点和谱点重合，并且对于\(\overline{\mathbb C}\)中的每个特征值，它们的约旦链之间存在双射。
英文摘要
In this paper $S$ is a closed symmetric linear relation in a Krein space $\mathfrak H$ with adjoint $S^*$, finite and equal defect numbers $d$, and a boundary mapping $\mathsf{b}: S^* \rightarrow \mathbb{C}^{2d}$ with Gram matrix $\mathsf Q$. We introduce a class $\mathbb A_{S,\mathsf{b}}$ of self-adjoint extensions of $S$ in a Krein space $\widetilde{\mathfrak H}$ containing $\mathfrak H$ as a Krein subspace of finite codimension, together with a class $\mathbb{P}_{\mathsf Q}$ of $d \times 2d$ matrix polynomials. Both classes are equipped with natural equivalence relations. Given $\mathcal{P}(z) \in \mathbb{P}_{\mathsf Q}$, we consider a boundary eigenvalue problem defined by the condition $\mathcal{P}(z)\mathsf{b}(\{f,g\})=0$, $\{f,g\}\in S^*$, and look for its linearizations. By a linearization we mean a linear relation $\widetilde{A} \in \mathbb{A}_{S, \mathsf b}$ such that the Shtraus extension $T_{\widetilde A}(z)$ of $S$ determined by $\widetilde A$ coincides, for all $z \in \overline{\mathbb C}$, with the linear relation $\big\{\{f,g\}\in S^*\,:\,\mathcal P(z)\mathsf b(\{f,g\})=0\big\}$. We prove that this correspondence defines a bijection between the equivalence classes in $\mathbb A_{S, \mathsf b}$ and those in $\mathbb P_{\mathsf Q}$. Moreover, we provide a condition under which the resulting linearizations are spectrally equivalent to the boundary eigenvalue problem: their regular and spectral points coincide, and for each eigenvalue in $\overline{\mathbb C}$ there is a bijection between their Jordan chains.
