Some properties of third order differential operators

Consider the third order differential operator L given by L(.) = 1/a(3)(t) d/dt 1/a(2)(t) d/dt 1/a(1)(t) d/dt (.) and the related linear differential equation L(x)(t) + x(t) = 0. We study the relations between L, its adjoint operator, the canonical representation of L, the operator obtained by a cyclic permutation of coefficients a(i), i = 1, 2, 3, in L and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).