We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-) symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-orthogonal subspaces. By orthogonality preserving mappings, these characterizations can be transferred to abstract boundary value spaces of (skew-)symmetric operators. Introducing the notion of boundary systems we then present a unified treatment of different versions of boundary triples and related concepts treated in the literature. The application of the abstract results yields a description of all (skew-)self-adjoint realizations of Laplace and first derivative operators on graphs.
- (skew)-self-adjoint operators
- quantum graphs
- boundary triple
Schubert, C., Seifert, C., Voigt, J., & Waurick, M. (2015). Boundary systems and (skew-)self-adjoint operators on infinite metric graphs. Mathematische Nachrichten, 288(14-15), 1776-1785. https://doi.org/10.1002/mana.201500054