  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
operators can be written as the sum of a skew-selfadjoint 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.
