Boundary systems and (skew-)self-adjoint operators on infinite metric graphs

TL;DR

This paper develops a unified framework for characterizing self-adjoint and skew-self-adjoint operators on infinite metric graphs using boundary systems and generalized Lagrangian subspaces, extending boundary triple theory.

Contribution

It introduces boundary systems and extends the concept of Lagrangian subspaces to characterize operators, unifying various boundary value approaches on graphs.

Findings

Provides a comprehensive description of self-adjoint realizations of Laplace operators on graphs.

Unifies different boundary triple concepts through boundary systems.

Characterizes operators via self-orthogonal subspaces with respect to symmetric forms.

Abstract

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-ortho-gonal 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.