On the similarity of boundary triples for dual pairs
Rytis Jursenas
TL;DR
This work analyzes how the Weyl family $M_{\Gamma^B}(\lambda)$ of a dual pair $(A,B)$ in a Krein space encodes the associated boundary triple $\Pi=(\mathrsfso{G},\Gamma^B,\Gamma^A)$ and, in particular, when this encoding determines the triple up to similarity and when that similarity can be chosen unitary. It develops a framework linking dual-pair boundary triples to a block-diagonal symmetric relation $T=\bigl(A00B\bigr)$, proving that $M_{\Gamma^B}$ determines $\Pi$ up to similarity, and that for $A=B$ unitary similarity is characterized by a unitary relation condition $(\Gamma^B)^{-1}\Gamma^{B'}=(\Gamma^A)^{-1}\Gamma^{A'}$. The paper then presents two concrete illustrations: (i) D-boundary triples in $\Pi_\kappa$-spaces, where the Weyl function over a half-plane determines the triple up to unitary equivalence; and (ii) a construction based on fractional linear transformations of Nevanlinna functions, showing how a Weyl function $M_{\dot\Gamma}$ transforms under a boundary-map $W$ to yield $M_{\Gamma^B}(\lambda)=W M_{\dot\Gamma}(\lambda)$ and how unitary similarity can be analyzed via explicit data. Overall, the results strengthen the link between Weyl data and boundary triples for dual pairs, with implications for boundary-value problems in indefinite metric spaces and for reconstructing boundary conditions from spectral data.
Abstract
The Weyl family of a dual pair $A\subseteq B^c$ of operators in a Krein space determines a minimal boundary triple uniquely up to similarity; if $A=B$, a necessary and sufficient condition in order that the similarity should be unitary is given.