Weyl families of transformed boundary pairs

Rytis Juršėnas

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3411-3448

Abstract: Let (L,Γ)$(\mathfrak {L},\Gamma )$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space H$\mathfrak {H}$. Let MΓ$M_\Gamma$ be the Weyl family corresponding to (L,Γ)$(\mathfrak {L},\Gamma )$. We cope with two main topics. First, since MΓ$M_\Gamma$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation MΓ(z)$M_\Gamma (z)$, for some z∈C∖R$z\in \mathbb {C}\setminus \mathbb {R}$, becomes a nontrivial task. Regarding MΓ(z)$M_\Gamma (z)$ as the (Shmul'yan) transform of zI$zI$ induced by Γ, we give conditions for the equality in MΓ(z)¯⊆MΓ¯(z)¯$\overline{M_\Gamma (z)}\subseteq \overline{M_{\overline{\Gamma }}(z)}$ to hold and we compute the adjoint MΓ¯(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 MΓ(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 MΓ$M_\Gamma$ corresponding to a boundary relation Γ 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′,Γ′)$(\mathfrak {L}^\prime ,\Gamma ^\prime )$ with its Weyl family MΓ′$M_{\Gamma ^\prime }$. The transformation scheme is either Γ′=ΓV−1$\Gamma ^\prime =\Gamma V^{-1}$ or Γ′=VΓ$\Gamma ^\prime =V\Gamma$ with suitable linear relations V. Results in this direction include but are not limited to: a 1‐1 correspondence between (L,Γ)$(\mathfrak {L},\Gamma )$ and (L′,Γ′)$(\mathfrak {L}^\prime ,\Gamma ^\prime )$; the formula for MΓ′−MΓ$M_{\Gamma ^\prime }-M_\Gamma$, 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,Γ0,Γ1)$(\mathfrak {L},\Gamma _0,\Gamma _1)$ with kerΓ=T$\ker \Gamma =T$ and T0=T0∗$T_0=T^*_0$ (second scheme, Hilbert space case).

Date: 2023
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