变换边界对的Weyl族
Weyl families of transformed boundary pairs
摘要：
设$(\mathfrak{L},Γ)$是Krein空间$\mathfrak{H}$中与闭对称线性关系$T$相关联的等距边界对。令$M_Γ$为对应于$(\mathfrak{L},Γ)$的Weyl族。我们处理两个主要议题。首先，由于$M_Γ$不需要（广义）Nevanlinna，封闭和伴随的线性关系$M_Γ(z)$的特征，对于某些$z\in\mathbb{C}\smallsetminus\mathbb{R}$，成为一个非平凡的任务。把$M_Γ(z)$看作是由$Γ$诱导的$zI$的（Shmul'yan）变换，给出了$\overline{M_Γ(z)}\subseteq\overline{M_{\overlineΓ}(z)}$中等式成立的条件，并计算了伴随$M_{\overlineΓ}(z)^*$ .作为一个应用程序，我们问当主变换的预解集与一个单一的边界对$T^+$是非空的。基于$M_Γ(z)$的封闭性准则，我们给出了答案的一个充分条件。例如，如果$T$是庞特里亚金空间中的标准线性关系，则对应于$T^+$的边界关系$Γ$的Weyl族$M_Γ$是广义Nevanlinna族。在第二个主题中，我们描述了转换后的边界对$(\mathfrak{L}^\prime,Γ^\prime)$与它的Weyl家族$M_{Γ^\prime}$。变换方案是具有适当线性关系$V$的$Γ^\prime=ΓV^{-1}$或$Γ^\prime=VΓ$。在这个方向上的结果包括但不限于：$(\mathfrak{L},Γ)$和$(\mathfrak{L}^\prime,Γ^\prime)$之间的1-1对应;对于普通边界三元组和标准酉算子$V$，$M_{Γ^\prime}-M_Γ$的公式;从等距边界三元组$(\mathfrak{L},Γ_0,Γ_1)$用$\kerΓ=T$和$T_0=T^*_0$构造准边界三元组。
英文摘要：
Let $(\mathfrak{L},Γ)$ be an isometric boundary pair associated with a closed symmetric linear relation $T$ in a Krein space $\mathfrak{H}$. Let $M_Γ$ be the Weyl family corresponding to $(\mathfrak{L},Γ)$. We cope with two main topics. First, since $M_Γ$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_Γ(z)$, for some $z\in\mathbb{C}\smallsetminus\mathbb{R}$, becomes a nontrivial task. Regarding $M_Γ(z)$ as the (Shmul'yan) transform of $zI$ induced by $Γ$, we give conditions for the equality in $\overline{M_Γ(z)}\subseteq\overline{M_{\overlineΓ}(z)}$ to hold and we compute the adjoint $M_{\overlineΓ}(z)^*$. As an application we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^+$ is nonempty. Based on the criterion for the closeness of $M_Γ(z)$ we give a sufficient condition for the answer. It follows, for example, that, if $T$ is a standard linear relation in a Pontryagin space then the Weyl family $M_Γ$ corresponding to a boundary relation $Γ$ for $T^+$ is a generalized Nevanlinna family. In the second topic we characterize the transformed boundary pair $(\mathfrak{L}^\prime,Γ^\prime)$ with its Weyl family $M_{Γ^\prime}$. The transformation scheme is either $Γ^\prime=ΓV^{-1}$ or $Γ^\prime=VΓ$ with suitable linear relations $V$. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},Γ)$ and $(\mathfrak{L}^\prime,Γ^\prime)$; the formula for $M_{Γ^\prime}-M_Γ$, for an ordinary boundary triple and a standard unitary operator $V$; construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},Γ_0,Γ_1)$ with $\kerΓ=T$ and $T_0=T^*_0$.
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