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

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