Classification of classical Friedrichs differential operators: One-dimensional scalar case

TL;DR

This paper advances the theoretical understanding of one-dimensional scalar Friedrichs operators by decomposing their graph space and classifying boundary conditions, building on recent operator-theoretic frameworks.

Contribution

It develops a graph space decomposition and provides a complete classification of boundary conditions for classical Friedrichs operators in one dimension.

Findings

Decomposition of the graph space into minimal domain and kernel components.

Complete classification of admissible boundary conditions.

Extension of operator-theoretic descriptions to classical scalar cases.

Abstract

The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems of first order partial differential equations (Friedrichs, 1958), nowadays better known as Friedrichs systems. Recently, Antoni\'c, Michelangeli and Erceg (2017) presented a purely operator-theoretic description of abstract Friedrichs operators, allowing for application of the universal operator extension theory (Grubb, 1968). In this paper we make a further theoretical step by developing a decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints. We then study one-dimensional scalar (classical) Friedrichs operators with variable coefficients and present a complete classification of admissible boundary conditions.