The von Neumann extension theory for abstract Friedrichs operators

The theory of abstract Friedrichs operators was introduced some fifteen years ago with the aim of providing a more comprehensive framework for the study of positive symmetric systems of first-order partial differential equations, nowadays better known as (classical) Friedrichs systems. Since then, the theory has not only been frequently applied in numerical and analytical research of Friedrichs systems, but has continued to evolve as well. In this paper, we provide an explicit characterisation and a classification of abstract Friedrichs operators. More precisely, we show that every abstract Friedrichs operator can be written as the sum of a skew-symmetric operator and a bounded self-adjoint strictly positive operator. Furthermore, we develop a classification of realisations of abstract Friedrichs operators in the spirit of the von Neumann extension theory, which, when applied to the symmetric case, extends the classical theory.