Uniform convergence of ergodic limits and approximate solutions
Authors: Shaw Sen-Yen 
Published: June 2010

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper A Subscript alpha Baseline StartAbsoluteValue EndAbsoluteValue equals upper O left-parenthesis 1 right-parenthesis comma upper A Subscript alpha Baseline upper A subset-of upper A upper A Subscript alpha Baseline comma StartAbsoluteValue EndAbsoluteValue upper A upper A Subscript alpha Baseline StartAbsoluteValue EndAbsoluteValue equals o left-parenthesis 1 right-parenthesis">

|

|

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-parenthesis upper P right-parenthesis equals upper N left-parenthesis upper A right-parenthesis circled-plus ModifyingAbove upper R left-parenthesis upper A right-parenthesis With bar comma upper R left-parenthesis upper P right-parenthesis equals upper N left-parenthesis upper A right-parenthesis comma upper D left-parenthesis upper B right-parenthesis equals upper A left-parenthesis upper D left-parenthesis upper A right-parenthesis intersection ModifyingAbove upper R left-parenthesis upper A right-parenthesis With bar right-parenthesis comma upper R left-parenthesis upper B right-parenthesis equals upper D left-parenthesis upper A right-parenthesis intersection ModifyingAbove upper R left-parenthesis upper A right-parenthesis With bar">

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.