论文标题
转化边界对的Weyl家族
Weyl families of transformed boundary pairs
论文作者
论文摘要
令$（\ mathfrak {l}，γ）$为与封闭的对称线性关系$ t $相关联的等距边界对。令$m_γ$为对应于$（\ mathfrak {l}，γ）$的Weyl家族。我们应对两个主要主题。首先，由于$m_γ$不必（概括）nevanlinna，因此对于某些$ z \ in \ mathbb {c} \ smalleStminus \ mathbb {r} $而言，闭合和线性关系$m_γ（z）$的伴随表征，成为一项非琐事。关于$m_γ（z）$作为$γ$引起的$ zi $的（shmul'yan）转换，我们给出了$ \ overline {m_γ（z）} \ subseteq \ subseteq \ subseteq \ overline {m_ _ {\overlineγ}（z）} $的条件，以固定和unagepline $ m____________ \ n exectaute nline {\overlineγ}（z）} $。作为一个应用程序，我们询问何时与$ t^+$的单一边界对关联的主要变换的分解集是非空的。根据$M_γ（z）$接近的标准，我们为答案提供了足够的条件。例如，例如，如果$ t $是Pontryagin空间中的标准线性关系，那么Weyl家族$M_γ$对应于边界关系$γ$ for $ t^+$是一个广义的Nevanlinna家族。在第二个主题中，我们表征了转换的边界对$（\ mathfrak {l}^\ prime，γ^\ prime）$，其weyl family $ m_ {γ^\ prime} $。转换方案是$γ^\ prime =γv^{ - 1} $或$γ^\ prime =vγ$，带有合适的线性关系$ v $。导致该方向包括但不限于以下方向：$（\ mathfrak {l}，γ）$和$（\ Mathfrak {l}^\ prime，γ^\ prime）$之间的1-1对应关系； $ m_ {γ^\ prime}-m_γ$的公式，用于普通边界三重和标准统一操作员$ v $;从等距边界三重$（\ mathfrak {l}，γ_0，γ_1）$构建准边界三重，并用$ \kerγ= t $和$ t_0 = t_0 = t^*_ 0 $。
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 $T_0=T^*_0$.
