If
P
(
x
,
∂
)
P(x,\partial )
is an
r
×
r
r\times r
system of differential operators on
R
N
\mathbb {R}^{N}
having continuous coefficients with vanishing oscillation at infinity, the Cordes–Illner theory ensures that
P
(
x
,
∂
)
P(x,\partial )
is Fredholm from
(
W
m
,
p
)
r
(W^{m,p})^{r}
to
(
L
p
)
r
(L^{p})^{r}
for all or no value
p
∈
(
1
,
∞
)
.
p\in (1,\infty ).
We prove that both the index (when defined) and the spectrum of
P
(
x
,
∂
)
P(x,\partial )
are independent of
p
.
p.