R E S E A R C H
Open Access
On conditional mean ergodic semigroups of
random linear operators
Xia Zhang
**Correspondence:
[email protected] Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin 300160, People’s Republic of China
Abstract
In this article, we prove two forms of conditional mean ergodic theorem for a strongly continuous semigroup of random isometric linear operators generated by a
semigroup of measure-preserving measurable isomorphisms, one of which generalizes and improves several known important results.
1 Introduction and the main results
The notion of a random normed module (briefly, anRNmodule), which was first intro-duced in [] and subsequently elaborated in [], is a random generalization of that of a normed space. In the last years, the theory ofRNmodules together with their random conjugate spaces have undergone a systematic and deep development [–], in particular the random reflexivity based on the theory of random conjugate spaces and the study of semigroups of random linear operators have also obtained some substantial advances in [, , –].
It is well known that the (ε,λ)-topology induced by theL-norm on anRN module is
exactly the topology of convergence in probabilityP. Actually, it is Mustari and Taylor that earlier observed the essence of the (ε,λ)-topology, studied probability theory in Banach spaces and did many excellent works [, ] under the framework of the specialRN mod-uleL(E,X), whereL(E,X) is theRNmodule of equivalence classes ofX-valued random
variables defined on a probability space (,E,P), see [] or Section for the construction ofL(E,X). Motivated by these works, we have recently begun to study the mean ergodic
theorem under the framework ofRNmodules to obtain the mean ergodic theorem in the sense of convergence in probability, in particular we proved a mean ergodic theorem for a strongly continuous semigroup of random unitary operators defined on complete random inner product modules in [] and further investigated the mean ergodicity for an almost surely bounded strongly continuous semigroup of random linear operators on a random reflexiveRNmodule in []. Based on these and motivated by the idea of [, ], the pur-pose of this article is to investigate the conditional mean ergodicity for a special semigroup of random linear operator on theRNmoduleLpF(E,X) and the construction ofLpF(E,X) is detailed as follows.
Now let us first recall the construction of LpF(E) in []. Let (,E,P) be a probabil-ity space,F a subσ-algebra ofE andL¯(E) (orL(E)) the set of equivalence classes of
E-measurable extended real-valued (real-valued) random variables on. LetL¯+(E) ={ξ∈
¯
L(E)|ξ≥}andL
+(E) ={ξ∈L(E)|ξ≥}. Similarly, one can understand such notions as
¯
L(F),L(F),L
+(F) andL¯+(F). Define the mapping| · |p:L¯(E)→ ¯L+(F) by |x|p=
E|x|p|Fp
for any x∈ ¯L(E) and ≤p<∞, whereE(|x|p|F) =lim
n→∞E(|x|p∧n|F) denotes the
extended conditional expectation, and let
LpF(E) =x∈L(E)||x|p∈L+(F)
,
then (LpF(E),| · |p) is anRN module. In fact,LpF(E) is exactly theL(F)-module
gen-erated by Lp(E), namely Lp
F(E) =L(F)·Lp(E) :={ξx|ξ ∈L(F) andx∈Lp(E)}, where
Lp(E) ={x∈L(E)|E[|x|p] <∞}. Further, motivated by the idea of constructingLp
F(E), Guo
in [] constructed a more generalRNmoduleLpF(S) and established the random conju-gate representation theorem of this type ofRNmodule. Precisely speaking, let (S, · ) be anRNmodule over the scalar fieldKwith base (,E,P), the mapping| · |p:S→ ¯L+(F)
is defined as follows for anyx∈Sandp∈[,∞):
|x|p=
Exp|F
p,
whereE(xp|F) =limn→∞E(xp∧n|F) also denotes the extended conditional
expecta-tion and let
LpF(S) =x∈S||x|p∈L+(F)
.
Then (LpF(S),| · |p) is anRNmodule overKwith base (,F,P). If we takeS=L(E), then
LpF(S) is exactlyLpF(E); if we further takeS=L(E,X), thenLpF(L(E,X)) (briefly,LpF(E,X)) is anRNmodule overKwith base (,F,P).
We can now state the main results of this article as follows.
Theorem . Let (,E,P) be a probability space, X a reflexive Banach space over K ,
{ht:t≥}a semigroup of measure-preserving measurable isomorphisms on(,E,P), p
an arbitrary positive number such that <p<∞ andF ={A∈E|h–
t (A) =A,∀t≥}.
Iflimt→|x◦ht –x|p= ,∀x∈LpF(E,X), then, for any x∈LpF(E,X), there exists some
¯
x∈LpF(E,X)such thatx¯=x¯◦htfor each t≥and
lim
r→∞
r
r
(x◦ht)dt=x¯ ()
in the(ε,λ)-topology induced by| · |p.
Theorem . Let(,E,P), X,{ht :t≥}andF be the same as in Theorem . andp¯
a given positive number such that <p¯<∞andlimt→|x◦ht–x|p¯ = ,∀x∈LpF¯ (E,X). Then, for any x∈L
F(E,X), there exists somex¯∈LF(E,X)such thatx¯=x¯◦htfor each t≥
and
lim
r→∞
r
r
(x◦ht)dt=x¯ ()
The remainder of this article is organized as follows: in Section we briefly recall some necessary basic notions and facts and Section is devoted to the proof of our main results.
2 Preliminaries
In the sequel of this article, (,E,P) denotes a probability space,K the scalar fieldRof real numbers orCof complex numbers, andL(E,K) the algebra of equivalence classes of
K-valuedE-measurable random variables onunder the ordinary addition, scalar mul-tiplication and mulmul-tiplication operations on equivalence classes. Besides, letL(E),L¯(E)
andL
+(E) be the same as in Section .
It is well known from [] thatL¯(E) is a complete lattice under the ordering≤:ξ≤η
if and only if ξ(ω)≤η(ω) forP-almost allωin(briefly, a.s.), where ξ andηare arbitrarily chosen representatives ofξandη, respectively. Furthermore, every subsetAof
¯
L(E) has a supremum, denoted by A, and an infimum, denoted byA, and there exist two sequences{an,n∈N}and{bn,n∈N}inAsuch that n≥an=∨Aand
n≥bn=
A. Finally,L(E), as a sublattice ofL¯(E), is complete in the sense that every subset with an
upper bound has a supremum.
Definition .[, ] An ordered pair (S, · ) is called anRNmodule overK with base (,E,P) ifSis a left module over the algebraL(E,K) and · is a mapping fromSto L
+(E) such that the following three axioms are satisfied: () ξx=|ξ|x,∀ξ∈L(E,K)andx∈S;
() x+y ≤ x+y,∀x,y∈S;
() x= impliesx=θ (the null vector ofS), where · is called theL-norm onS andxis called theL-norm of a vectorx∈S.
It should be pointed out that the following idea of introducing the (ε,λ)-topology is due to Schweizer and Sklar [].
Let (S, · ) be anRNmodule overKwith base (,E,P). For any positive real numbers εandλsuch thatλ< , letNθ(ε,λ) ={x∈S|P{ω∈|x(ω) <ε}>λ}, then{Nθ(ε,λ)|ε> , <λ< }is a local base at the null vectorθ of some Hausdorff linear topology. The linear topology is called the (ε,λ)-topology. In this article, given anRNmodule (S, · ) overK with base (,E,P), it is always assumed that (S, · ) is endowed with the (ε, λ)-topology. One only needs to notice that a sequence{xn,n∈N}inSconverges tox∈Sin
the (ε,λ)-topology if and only if{xn–x,n∈N}converges to in probabilityP.
Example Let X be a normed space over K and L(E,X) the linear space of
equiva-lence classes ofX-valuedE-random variables on. The module multiplication opera-tion·:L(E,K)×L(E,X)→L(E,X) is defined byξx= the equivalence class ofξx,
whereξandxare the respective arbitrarily chosen representatives ofξ∈L(E,K) and x∈L(E,X), and (ξx)(ω) =ξ(ω)·x(ω), ∀ω∈. Furthermore, the mapping · : L(E,X)→L
+(E) byx= the equivalence class ofx, ∀x∈L(E,X), wherex is as
above. Then it is easy to see that (L(E,X),·) is anRNmodule overKwith base (,E,P).
Definition .[] Let (S, ·
) and (S, · ) be twoRN modules overK with base
random linear operators fromS toS, define · :B(S,S)→L
+(E) byT:=
{ξ ∈ L
+(E)|Tx≤ξ· x,∀x∈S}for anyT ∈B(S,S), then it is easy to see that (B(S,S), · ) is anRNmodule overKwith base (,E,P).
Specially, denote (B(S,S), · ) by (S*, · *) when (S, ·
) is a givenRN module
(S, · ) overKwith base (,E,P) andS=L(E,K), then (S*, · *) is called the random
conjugate space of (S, · ). Let (S**, · **) be the random conjugate space of (S*, · *).
The canonical embedding mappingJ:S→S**defined by (Jx)(f) =f(x) for anyx∈Sand
f ∈S*, is random-norm preserving. IfJis surjective, thenSis called random reflexive [].
Definition .[, ] Let (S, · ) be anRNmodule overKwith base (,E,P),B(S) the set of a.s. bounded random linear operators onS. A family{B(t) :t≥} ⊂B(S) is called a semigroup of random linear operators if
B() =I and B(s)B(t) =B(s+t)
for alls,t≥, whereIdenotes the identity operator onS. Further, if the mappingB(·)x: [, +∞)→S,t→B(t)xis continuous w.r.t. the (ε,λ)-topology for everyx∈S, then the semigroup of random linear operators{B(t) :t≥}is said to be strongly continuous. Be-sides, if t≥B(t) ∈L
+(E), then{B(t) :t≥}is called an a.s. bounded strongly
contin-uous semigroup of random linear operators.
Proposition .[] Let(S, ·
)and(S, · )be two RN modules over K with base
(,E,P). Then we have the following statements:
() T∈B(S,S)if and only ifTis a continuous module homomorphism;
() IfT∈B(S,S), thenT=∨{Tx:x∈Sandx≤}, wheredenotes the identity element inL(E).
3 Proof of the main results
The proof of Theorem . needs Theorem . and Lemma . below. To prove Theo-rem . and introduce Lemma ., we will first recall the definition of Riemann integral for abstract-valued functions from a finite real interval to anRNmodule and a sufficient condition for such a function to be Riemann-integrable.
Let [a,b] be a finite real interval andP={x,x, . . . ,xk, . . . ,xn}a finite partition into [a,b],
namely,a=x<x<· · ·<xn–<xn=bandλ(P) =max≤i≤n{xi}, wherexi=xi–xi–(i=
, . . . ,n). Besides, in the following of this section we always suppose that (S, · ) denotes a completeRNmodule overKwith base (,E,P).
Definition .[] Letf be a function from [a,b] toS.f is called Riemann-integrable on [a,b] if there exists someIinSwith the following property: for any positive numbersε andλwithλ< there is a positive numberδ(ε,λ) such that
P
ω∈
n
i=
f(ξi)xi–I
(ω) <ε
> –λ
for any finite partitionP={x,x, . . . ,xk, . . . ,xn}and arbitrarily chosenξi∈[xi–,xi] (≤
Proposition . [] Let f be a continuous function from [a,b] to S such that
t∈[a,b]f(t) ∈L+(E), then f is Riemann integrable in the(ε,λ)-topology on[a,b].
Fur-ther, if t∈[a,b]f(t) ∈L(,E,R), then f is Riemann integrable in ·
on[a,b], where · denotes the -norm of the Banach space L(,E,R).
Definition .[] Let{B(t) :t≥}be an a.s. bounded strongly continuous semigroup
of random linear operators on anRNmoduleS. We denote by
C(r)x:= r
r
B(s)x ds, ∀x∈S,r>
the Cesàro means of{B(t) :t≥}. For anyx∈S, if{C(r)x,r> }converges to some point
inSasr→ ∞, then{B(t) :t≥}is called mean ergodic.
Note that it is Proposition . that makes Definition . be well defined.
Proposition .[] Let(S, · )be a complete RN module over K with base(,E,P). If
S is random reflexive, then every a.s. bounded strongly continuous semigroup of random linear operators on S is mean ergodic.
It is known from [] thatL(E,X) is random reflexive if and only ifXis a reflexive Banach
space. Besides, Guo [] proved that if anRNmoduleSis random reflexive, thenLpF(S) is also random reflexive. Based on these facts as well as the preceding Proposition ., we
can now prove Theorem . as follows.
Proof of Theorem . For eacht≥, define the mappingT(t) :LpF(E,X)→LpF(E,X) by T(t)x=x◦htfor eachx∈LpF(E,X), then it is clear thatT(t) is a module homomorphism.
Furthermore,
T()x=x◦h=x
for anyx∈LpF(E,X) and
T(s)T(t)x= (x◦ht)◦hs=x◦ht+s=T(s+t)x
SinceE(T(t)xp∧n|F),E(xp∧n|F)∈L(,F,P) for eacht≥,x∈Lp
F(E,X) and
n∈N, it follows that
A
ET(t)xp∧nF(ω)P(dω)
=
A
T(t)xp
(ω)∧nP(dω)
=
A
xht(ω) p
∧nP(dω)
=
h–t (A)
xht(ω)p∧n
P(dω)
=
A
x(ω)p
∧nP(dω)
=
A
Exp∧n|F(ω)P(dω)
()
for anyA∈F. Thus we have
ET(t)xp∧n|F=Exp∧n|F ()
for anyt≥,x∈LpF(E,X) andn∈N, lettingn→ ∞in () yields that
ET(t)xp|F=Exp|F,
namely|T(t)x|pp=|x|ppfor anyx∈LpF(E,X), which shows that
T(t) :LpF(E,X),| · |p
→LpF(E,X),| · |p
is a random isometric linear operator for eacht≥. Moreover,
lim
t→|x◦ht–x|p= , ∀x∈L
p
F(E,X).
Thus{T(t) :t≥}is a strongly continuous semigroup of random isometric linear opera-tors onLpF(E,X).
Clearly,{T(t) :t≥}is an a.s. bounded strongly continuous semigroup of random lin-ear operators onLpF(E,X). SinceXis reflexive, it follows that theRN moduleL(E,X)
is random reflexive, further,LpF(E,X) is random reflexive. Thus it follows from Proposi-tion . thatT(t) is mean ergodic for eacht≥, i.e. for anyx∈LpF(E,X) there exists some
¯
x∈LpF(E,X) such that
lim
r→∞ r
r
(x◦ht)dt= lim r→∞
r
r
T(t)x dt=x.¯
Now it remains to show thatx¯=x¯◦htfor eacht≥. In fact, since
T(s)
r
r
T(t)x dt
= r
r
T(s+t)x dt= r
r+s s
and
r
r+s s
T(t)x dt
= r
r+s
T(t)x dt– r
s
T(t)x dt
=r+s r
r+s
r+s
T(t)x dt– r
s
T(t)x dt,
()
for anyr,s> andx∈S, fixsandx, lettingr→ ∞in () yields
lim
r→∞T(s)
r
r+s s
T(t)x dt
=x.¯
Further,T(t) is a random isometric linear operator for eacht≥, thus we haveT(t)x¯=x,¯ namelyx¯=x¯◦ht.
This completes the proof.
It should be pointed out that Lemma . is merely mentioned in [] and partially proved in [].
Lemma . Let f be a continuous function from[a,b]to L(E)such that
t∈[a,b]|f(t)| ∈
L+(E). Then
b a
f(t)dt
dP= b a f(t)dP
dt. ()
Proof Letξ= t∈[a,b]|f(t)|, thenξ∈L
+(E). LetHm= [m– ≤ξ<m] for eachm∈N, then
{Hm,m∈N}is a sequence of pairwise disjointF-measurable sets such that
∞
m=Hm=
. Define a mapping fm : [a,b]→S by fm(t) =IHm ·f(t) for each m∈N. Obviously,
t∈[a,b]|fm(t)| ∈L(,E,R) for eachm∈Nand eacht∈[a,b], thus it follows by
Proposi-tion . thatfmis Riemann integrable in · on [a,b]. LetP,λ(P) andxi,ξi(≤i≤n)
be the same as in Definition . andIm=
b
a fm(t)dt,Imn=
n
i=fm(ξi)xifor eachm∈N.
Since
ImndP–
ImdP
≤
|Imn–Im|dP
≤
|Imn–Im|dP
=Imn–Im
()
for each fixedm∈N, we have
ImdP= lim
λ(P)→
ImndP= lim
λ(P)→
n
i=
fm(ξi)dPxi=
b a
fm(t)dP
Thus for anyk∈N, we have k m= b a
fm(t)dt
dP= k m= b a fm(t)dP
dt. () Since ∞ m=
P(Hm) =P
∞ m= Hm =P(),
it follows that {km=[abfm(t)dt]dP,k∈N} and{
k m=
b a[
fm(t)dP]dt,k∈N} are both Cauchy sequences. Lettingk→ ∞in Equation () yields Equation ().
We can now prove Theorem ..
Proof of Theorem . LetT(t)x=x◦ht for eacht≥ andx∈LpF¯ (E,X), then it follows
from Theorem . that
r
r
(x◦ht)dt→ ¯x (n→ ∞)
in the (ε,λ)-topology induced by| · |p¯. Thus by the random Schwarz-Cauchy inequality, we have
rr(x◦ht)dt–x¯
→ (n→ ∞)
in probabilityP. For anyx∈L
F(E,X), since
r r
(x◦ht)dt
= r r
T(t)x dt
≤ |x|
for each n∈N andLpF¯ (E,X) is dense inL
F(E,X) in| · |, one can obtain the desired
Equation ().
Ifx∈L(,E,X), it suffices to show thatx¯=E(x|F). LetA∈F, then for eachn∈N, it
follows from Lemma . that
A r r
T(t)x dt
dP= r
r
A
T(t)x dP
dt = r r A
xht(ω)
P(dω) dt = r r
h–t (A)
xht(ω)
P(dω)
dt sinceh–t (A) =A ()
Furthermore, since
lim
n→∞
r
r
T(t)x dt=x¯
in the (ε,λ)-topology induced by| · |and
r r
(x◦ht)dt
≤ |x|
for eachx∈L(,E,X)⊂L
F(E,X), lettingr→ ∞in (), it follows from the Lebesgue’s
dominated convergence theorem that
A
¯
x(ω)P(dω) =
A
x(ω)P(dω),
namely the desired result follows.
LetHbe a Hilbert space overK. It is clear that Theorem . still holds whenXis taken place ofH, which includes the following known result.
Corollary .[] Let(,E,P)be a probability space,{ht:t≥}a semigroup of
measure-preserving measurable isomorphisms on (,E,P), F ={A∈E|h–
t (A) =A,∀t≥} and
limt→|x◦ht –x| = ,∀x∈LF(E,H). Then, for any x∈LF(E,H), there exists some ¯
x∈LF(E,H)such thatx¯=x¯◦htfor each t≥and
lim
r→∞ r
r
(x◦ht)dt=x.¯ ()
Corollary . If, in addition to the hypothesis of Theorem ., we assume that A∈F if and only if P(A) = of, thenx¯=E(x).
Competing interests
The author declare that they have no competing interests.
Acknowledgement
The author would like to express his sincere gratitude to Prof. Guo Tiexin for his invaluable suggestions. The study was supported by the National Natural Science Foundation of China (No. 11171015). The author would also like to thank the referees for their invaluable suggestions.
Received: 7 February 2012 Accepted: 8 May 2012 Published: 3 July 2012 References
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