Weyl families of transformed boundary pairs

Let ( L , Γ ) $(\mathfrak {L},\Gamma )$ be an isometric boundary pair associated with a closed symmetric linear relation T in Krein space H $\mathfrak {H}$ . M $M_\Gamma$ the Weyl family corresponding to We cope two main topics. First, since need not (generalized) Nevanlinna, characterization of closure and adjoint z $M_\Gamma (z)$ for some ∈ C ∖ R $z\in \mathbb {C}\setminus {R}$ becomes nontrivial task. Regarding as (Shmul'yan) transform I $zI$ induced by Γ, we give conditions equality ¯ ⊆ $\overline{M_\Gamma (z)}\subseteq \overline{M_{\overline{\Gamma }}(z)}$ hold compute ∗ $M_{\overline{\Gamma }}(z)^*$ As application, ask when resolvent set unitary + $T^+$ is nonempty. Based on criterion closeness sufficient condition answer. From this result it follows, example, that, if standard Pontryagin space, then generalized Nevanlinna family; similar conclusion already known operator. In second topic, characterize transformed ′ {L}^\prime ,\Gamma ^\prime its $M_{\Gamma }$ The transformation scheme either = V − 1 $\Gamma =\Gamma V^{-1}$ or =V\Gamma$ suitable relations V. Results direction include but are limited to: 1-1 correspondence between ; formula }-M_\Gamma$ ordinary triple operator (first scheme); construction quasi from 0 _0,\Gamma _1)$ ker $\ker \Gamma =T$ $T_0=T^*_0$ (second scheme, Hilbert case).