Let
K
(
s
,
t
)
K(s,t)
be a complex-valued
L
2
{L_2}
kernel on the square
a
≦
s
,
t
≦
b
a \leqq s,t \leqq b
and
{
λ
v
}
\{ {\lambda _v}\}
, perhaps empty, denote the set of finite characteristic values (f.c.v.) of
K
K
, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation
ϕ
v
(
s
)
=
λ
v
∫
a
b
K
(
s
,
t
)
ϕ
v
(
t
)
d
t
{\phi _v}(s) = {\lambda _v}\int _a^b {K(s,t){\phi _v}(t)dt}
, where the
ϕ
v
(
s
)
{\phi _v}(s)
are nontrivial
L
2
{L_2}
functions on
[
a
,
b
]
[a,b]
. Further let
k
1
=
∫
a
b
K
(
s
,
s
)
{k_1} = \int _a^b {K(s,s)}
be well defined so that the Fredholm determinant of
K
,
D
(
λ
)
K,D(\lambda )
, exists, and let
μ
\mu
be the order of this entire function. It is shown that (1) if
K
(
s
,
t
)
K(s,t)
is a function of bounded variation in the sense of Hardy-Krause, then
μ
≦
1
\mu \leqq 1
; (2) if in addition to the assumption (1),
K
(
s
,
t
)
K(s,t)
satisfies a uniform Lipschitz condition of order
α
>
0
\alpha > 0
with respect to either variable, then
μ
>
1
\mu > 1
and
k
1
=
Σ
v
1
/
λ
v
{k_1} = {\Sigma _v}1/{\lambda _v}
; (3) if
K
(
s
,
t
)
K(s,t)
is absolutely continuous as a function of two variables and
∂
2
K
/
∂
s
∂
t
{\partial ^2}K/\partial s\partial t
(which exists almost everywhere) belongs to class
L
p
{L_p}
for some
p
>
1
p > 1
, then
μ
>
1
\mu > 1
and
k
1
=
Σ
v
1
/
λ
v
{k_1} = {\Sigma _v}1/{\lambda _v}
. In (2) and (3), the condition
k
1
≠
0
{k_1} \ne 0
implies
K
(
s
,
t
)
K(s,t)
possesses at least one f.c.v.