On a boundary pair of a dissipative operator

TL;DR

The paper investigates how the boundary data of a dissipative operator $T$ in a Krein space is determined by the unitary boundary pair of its symmetric part. It constructs an intrinsic boundary space $\mathrsfso{E}$ as the completion of the positive $G$-space associated with the symmetric form $\gamma_T$, and expresses the boundary map as $\Gamma_{01}=V\Gamma_s|_{\mathrsfso{D}_T}$, linking $T$'s boundary data to that of $S^*$. A main result shows existence and canonical form of the boundary pair, and that all boundary pairs are unique up to a unitary transform on $\mathrsfso{E}$; the work also provides a practical completeness criterion for the associated form $\gamma_N$ and demonstrates applicability via a PDE example. These contributions clarify how symmetric-part boundary data governs dissipative extensions in indefinite inner product spaces, with implications for spectral analysis and operator model construction.

Abstract

The aim of this brief note is to demonstrate that the boundary pair of a dissipative operator is determined by the unitary boundary pair of its symmetric part.

Theorems & Definitions (15)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• Lemma 5
• proof
• Lemma 6
• proof
• Remark 8
• Proposition 9