Weyl families of transformed boundary pairs
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 $T_0=T^*_0$.