Ergodic projections of continuous and discrete semigroups
Authors: Shaw Sen-Yen 
Published: June 2010

Let X be a Banach space. Let

{
T
(
t
)
;
t
>
0
}

\{ T(t);t > 0\}

be a uniformly bounded semigroup of operators on X, which converges strongly to P, known to be a projection, as t goes to 0. If A is its generator and

X
0

{X_0}

[resp.,

X
t

,
t
>
0

{X_t},t > 0

] is the set of x for which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P 0 x identical-to limit Underscript t right-arrow normal infinity Endscripts t Superscript negative 1 Baseline integral Subscript 0 Superscript t Baseline upper T left-parenthesis tau right-parenthesis x d tau left-bracket resp period comma upper P Subscript t Baseline x identical-to limit Underscript n right-arrow normal infinity Endscripts n Superscript negative 1 Baseline sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts upper T left-parenthesis i t right-parenthesis x right-bracket">

P
0

x
≡

lim

t
→
∞

t

−
1

∫
0
t

T
(
τ
)
x

d
τ

[

resp

.
,

P
t

x
≡

lim

n
→
∞

n

−
1

∑

i
=
0

n
−
1

T
(
i
t
)
x

]

{P_0}x \equiv \lim \limits _{t \to \infty } {t^{ - 1}}\int _0^t {T(\tau )x\;d\tau \;\left [ {{\text {resp}}.,\;{P_t}x \equiv \lim \limits _{n \to \infty } {n^{ - 1}}\sum \limits _{i = 0}^{n - 1} {T(it)x} } \right ]}

\]
exists, then, for each

t
⩾
0
,

P
t

t \geqslant 0,{P_t}

is a bounded projection in

X
t

{X_t}

; when
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t equals 0 comma upper X 0 equals upper N left-parenthesis upper A right-parenthesis circled-plus ModifyingAbove upper R left-parenthesis upper A right-parenthesis With bar circled-plus upper N left-parenthesis upper P right-parenthesis comma upper R left-parenthesis upper P 0 right-parenthesis equals upper N left-parenthesis upper A right-parenthesis">

t
=
0
,

X
0

=
N
(
A
)
⊕

R
(
A
)

¯

⊕
N
(
P
)
,

R
(

P
0

)
=
N
(
A
)

t = 0,{X_0} = N(A) \oplus \overline {R(A)} \oplus N(P),\;R({P_0}) = N(A)

and

N
(

P
0

)
=

R
(
A
)

¯

⊕
N
(
P
)

N({P_0}) = \overline {R(A)} \oplus N(P)

; when

t
>
0

t > 0

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript t Baseline equals upper N left-parenthesis upper T left-parenthesis t right-parenthesis minus upper I right-parenthesis circled-plus ModifyingAbove upper R left-parenthesis upper T left-parenthesis t right-parenthesis minus upper I right-parenthesis With bar comma">

X
t

=
N
(
T
(
t
)
−
I
)
⊕

R
(
T
(
t
)
−
I
)

¯

,

{X_t} = N(T(t) - I) \oplus \overline {R(T(t) - I)} ,

\]

R
(

P
t

)
=
N
(
T
(
t
)
−
I
)

R({P_t}) = N(T(t) - I)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis upper P Subscript t Baseline right-parenthesis equals ModifyingAbove upper R left-parenthesis upper T left-parenthesis t right-parenthesis minus upper I right-parenthesis With bar semicolon upper X Subscript t Baseline equals upper X">

N
(

P
t

)
=

R
(
T
(
t
)
−
I
)

¯

;

X
t

=
X

N({P_t}) = \overline {R(T(t) - I)} ;\;{X_t} = X

for all

t
⩾
0

t \geqslant 0

if X is reflexive. Some results on relations among the projections

P
t

,
t
⩾
0

{P_t},t \geqslant 0

, are obtained. In particular, we have

P
t

=

P
0

{P_t} = {P_0}

for all sufficiently small t if A is bounded.