  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.
