The problem of factoring the general ordinary linear differential operator
L
y
=
y
(
n
)
+
p
n
−
1
y
(
n
−
1
)
+
⋯
+
p
0
y
Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y
into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation
L
y
=
0
Ly = 0
and its adjoint equation
L
∗
y
=
0
{L^ \ast }y = 0
. The special case when L is formally selfadjoint of order
n
=
2
k
n = 2k
and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz:
L
=
Q
∗
Q
L = {Q^ \ast }Q
if and only if there exist solutions
y
1
,
⋯
,
y
k
{y_1}, \cdots ,{y_k}
of
L
y
=
0
Ly = 0
satisfying
W
(
y
1
,
⋯
,
y
k
)
≠
0
W({y_1}, \cdots ,{y_k}) \ne 0
and
[
y
i
;
y
j
]
=
0
[{y_i};{y_j}] = 0
for
i
,
j
=
1
,
⋯
,
k
i,j = 1, \cdots ,k
; where [ ; ] is the Lagrange bilinear form of L.