Let
A
A
be a densely defined closed (linear) operator, and
{
A
α
}
\{ {A_\alpha }\}
,
{
B
α
}
\{ {B_\alpha }\}
be two nets of bounded operators on a Banach space
X
X
such that
|
|
A
α
|
|
=
O
(
1
)
,
A
α
A
⊂
A
A
α
,
|
|
A
A
α
|
|
=
o
(
1
)
||{A_\alpha }|| = O(1),{A_\alpha }A \subset A{A_\alpha },||A{A_\alpha }|| = o(1)
, and
B
α
A
⊂
A
B
α
=
I
−
A
α
{B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha }
. Denote the domain, range, and null space of an operator
T
T
by
D
(
T
)
D(T)
,
R
(
T
)
R(T)
, and
N
(
T
)
N(T)
, respectively, and let
P
(
resp
.
B
)
P(\operatorname {resp} .B)
be the operator defined by
P
x
=
lim
α
A
α
x
(
r
e
s
p
.
B
y
=
lim
α
B
α
y
)
Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y)
for all those
x
∈
X
(
resp
.
y
∈
R
(
A
)
¯
)
x \in X(\operatorname {resp} .y \in \overline {R(A)} )
for which the limit exists. It is shown in a previous paper that
D
(
P
)
=
N
(
A
)
⊕
R
(
A
)
¯
,
R
(
P
)
=
N
(
A
)
,
D
(
B
)
=
A
(
D
(
A
)
∩
R
(
A
)
¯
)
,
R
(
B
)
=
D
(
A
)
∩
R
(
A
)
¯
D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)}
, and that
B
B
sends each
y
∈
D
(
B
)
y \in D(B)
to the unique solution of
A
x
=
y
in
R
(
A
)
¯
Ax = y{\text { in }}\overline {R(A)}
. In this paper, we prove that
D
(
P
)
=
X
D(P) = X
and
|
|
A
α
−
P
|
|
→
0
||{A_\alpha } - P|| \to 0
if and only if
|
|
B
α
|
D
(
B
)
−
B
|
|
→
0
||{B_\alpha }|D(B) - B|| \to 0
, if and only if
|
|
B
α
|
D
(
B
)
|
|
=
O
(
1
)
||{B_\alpha }|D(B)|| = O(1)
, if and only if
R
(
A
)
R(A)
is closed. Moreover, when
X
X
is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that
D
(
P
)
=
X
D(P) = X
. The general result is then used to deduce uniform ergodic theorems for
n
n
-times integrated semigroups,
(
Y
)
(Y)
-semigroups, and cosine operator functions.