Some remarks on the notions of boundary systems and boundary triple(t)s

TL;DR

This paper explores the relationship between boundary systems and boundary triplets, showing how boundary systems can induce triplets and classify extensions, and provides explicit descriptions of maximal dissipative extensions for skew-symmetric operators.

Contribution

It establishes a connection between boundary systems and boundary triplets, and offers explicit descriptions of maximal dissipative extensions even when triplets do not exist.

Findings

Boundary systems can induce boundary triplets for skew-self-adjoint extensions.

The classification of extensions aligns with classical results when boundary triplets are present.

Every skew-symmetric operator admits a boundary system leading to explicit maximal dissipative extensions.

Abstract

In this note we show that if a boundary system in the sense of (Schubert et al. 2015) gives rise to any skew-self-adjoint extension, then it induces a boundary triplet and the classification of all extensions given by (Schubert et al. 2015) coincides with the skew-symmetric version of the classical characterization due to (Gorbachuk et al. 1991). On the other hand we show that for every skew-symmetric operator there is a natural boundary system which leads to an explicit description of at least one maximal dissipative extension. This is in particular also valid in the case that no boundary triplet exists for this operator.