Weyl families of transformed boundary pairs

Let (L,Gamma)$(\mathfrak {L},\Gamma )$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space H$\mathfrak {H}$. Let M Gamma$M_\Gamma$ be the Weyl family corresponding to (L,Gamma)$(\mathfrak {L},\Gamma )$. We cope with two main topics. First, since M Gamma$M_\Gamma$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation M Gamma(z)$M_\Gamma (z)$, for some z is an element of C set minus R$z\in \mathbb {C}\setminus \mathbb {R}$, becomes a nontrivial task. Regarding M Gamma(z)$M_\Gamma (z)$ as the (Shmul'yan) transform of zI$zI$ induced by Gamma, we give conditions for the equality in M Gamma(z) over bar subset of M Gamma over bar (z) over bar $\overline{M_\Gamma (z)}\subseteq \overline{M_{\overline{\Gamma }}(z)}$ to hold and we compute the adjoint M Gamma over bar (z)*$M_{\overline{\Gamma }}(z)<^>*$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for T+$T<^>+$ is nonempty. Based on the criterion for the closeness of M Gamma(z)$M_\Gamma (z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family M Gamma$M_\Gamma$ corresponding to a boundary relation Gamma for T+$T<^>+$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair (L ',Gamma ')$(\mathfrak {L}<^>\prime ,\Gamma <^>\prime )$ with its Weyl family M Gamma '$M_{\Gamma <^>\prime }$. The transformation scheme is either Gamma '=Gamma V-1$\Gamma <^>\prime =\Gamma V<^>{-1}$ or Gamma '=V Gamma$\Gamma <^>\prime =V\Gamma$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between (L,Gamma)$(\mathfrak {L},\Gamma )$ and (L ',Gamma ')$(\mathfrak {L}<^>\prime ,\Gamma <^>\prime )$; the formula for M Gamma '-M Gamma$M_{\Gamma <^>\prime }-M_\Gamma$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple (L,Gamma 0,Gamma 1)$(\mathfrak {L},\Gamma _0,\Gamma _1)$ with ker Gamma=T$\ker \Gamma =T$ and T0=T0*$T_0=T<^>*_0$ (second scheme, Hilbert space case).