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On conditional mean ergodic semigroups of random linear operators

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, an RN module), which was first introduced in [1] and subsequently elaborated in [2], is a random generalization of that of a normed space. In the last 10 years, the theory of RN modules together with their random conjugate spaces have undergone a systematic and deep development [39], 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 [6, 8, 1012].

It is well known that the (ε,λ)-topology induced by the L 0 -norm on an RN module is exactly the topology of convergence in probability P. 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 [13, 14] under the framework of the special RN module L 0 (E,X), where L 0 (E,X) is the RN module of equivalence classes of X-valued random variables defined on a probability space (Ω,E,P), see [4] or Section 2 for the construction of L 0 (E,X). Motivated by these works, we have recently begun to study the mean ergodic theorem under the framework of RN modules 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 [8] and further investigated the mean ergodicity for an almost surely bounded strongly continuous semigroup of random linear operators on a random reflexive RN module in [11]. Based on these and motivated by the idea of [15, 16], the purpose of this article is to investigate the conditional mean ergodicity for a special semigroup of random linear operator on the RN module L F p (E,X) and the construction of L F p (E,X) is detailed as follows.

Now let us first recall the construction of L F p (E) in [16]. Let (Ω,E,P) be a probability space, F a sub σ-algebra of E and L ¯ 0 (E) (or L 0 (E)) the set of equivalence classes of E-measurable extended real-valued (real-valued) random variables on Ω. Let L ¯ + 0 (E)={ξ L ¯ 0 (E)|ξ0} and L + 0 (E)={ξ L 0 (E)|ξ0}. Similarly, one can understand such notions as L ¯ 0 (F) L 0 (F) L + 0 (F) and L ¯ + 0 (F). Define the mapping p : L ¯ 0 (E) L ¯ + 0 (F) by

x p = [ E ( | x | p | F ) ] 1 p

for any x L ¯ 0 (E) and 1p<, where E( | x | p |F)= lim n E( | x | p n|F) denotes the extended conditional expectation, and let

L F p (E)= { x L 0 ( E ) | x p L + 0 ( F ) } ,

then ( L F p (E), p ) is an RN module. In fact, L F p (E) is exactly the L 0 (F)-module generated by L p (E), namely L F p (E)= L 0 (F) L p (E):={ξx|ξ L 0 (F)andx L p (E)}, where L p (E)={x L 0 (E)|E[ | x | p ]<}. Further, motivated by the idea of constructing L F p (E), Guo in [3] constructed a more general RN module L F p (S) and established the random conjugate representation theorem of this type of RN module. Precisely speaking, let (S,) be an RN module over the scalar field K with base (Ω,E,P), the mapping p :S L ¯ + 0 (F) is defined as follows for any xS and p[1,):

x p = [ E ( x p | F ) ] 1 p ,

where E( x p |F)= lim n E( x p n|F) also denotes the extended conditional expectation and let

L F p (S)= { x S | x p L + 0 ( F ) } .

Then ( L F p (S), p ) is an RN module over K with base (Ω,F,P). If we take S= L 0 (E), then L F p (S) is exactly L F p (E); if we further take S= L 0 (E,X), then L F p ( L 0 (E,X)) (briefly, L F p (E,X)) is an RN module over K with base (Ω,F,P).

We can now state the main results of this article as follows.

Theorem 1.1 Let(Ω,E,P)be a probability space, X a reflexive Banach space over K, { h t :t0}a semigroup of measure-preserving measurable isomorphisms on(Ω,E,P), p an arbitrary positive number such that1<p<andF={AE| h t 1 (A)=A,t0}. If lim t 0 x h t x p =0, x L F p (E,X), then, for anyx L F p (E,X), there exists some x ¯ L F p (E,X)such that x ¯ = x ¯ h t for eacht0and

lim r 1 r 0 r (x h t )dt= x ¯
(1)

in the(ε,λ)-topology induced by p .

Theorem 1.2 Let(Ω,E,P), X, { h t :t0}andFbe the same as in Theorem 1.1 and p ¯ a given positive number such that1< p ¯ <and lim t 0 x h t x p ¯ =0, x L F p ¯ (E,X). Then, for anyx L F 1 (E,X), there exists some x ¯ L F 1 (E,X)such that x ¯ = x ¯ h t for eacht0and

lim r 1 r 0 r (x h t )dt= x ¯
(2)

in the(ε,λ)-topology induced by 1 . In particular, ifx L 1 (Ω,E,X), then x ¯ =E(x|F).

The remainder of this article is organized as follows: in Section 2 we briefly recall some necessary basic notions and facts and Section 3 is devoted to the proof of our main results.

2 Preliminaries

In the sequel of this article, (Ω,E,P) denotes a probability space, K the scalar field R of real numbers or C of complex numbers, and L 0 (E,K) the algebra of equivalence classes of K-valued E-measurable random variables on Ω under the ordinary addition, scalar multiplication and multiplication operations on equivalence classes. Besides, let L 0 (E), L ¯ 0 (E) and L + 0 (E) be the same as in Section 1.

It is well known from [17] that L ¯ 0 (E) is a complete lattice under the ordering ≤: ξη if and only if ξ 0 (ω) η 0 (ω) for P-almost all ω in Ω (briefly, a.s.), where ξ 0 and η 0 are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset A of L ¯ 0 (E) has a supremum, denoted by A, and an infimum, denoted by A, and there exist two sequences { a n ,nN} and { b n ,nN} in A such that n 1 a n =A and n 1 b n =A. Finally, L 0 (E), as a sublattice of L ¯ 0 (E), is complete in the sense that every subset with an upper bound has a supremum.

Definition 2.1 [2, 3]

An ordered pair (S,) is called an RN module over K with base (Ω,E,P) if S is a left module over the algebra L 0 (E,K) and is a mapping from S to L + 0 (E) such that the following three axioms are satisfied:

  1. (1)

    ξx=|ξ|x, ξ L 0 (E,K) and xS;

  2. (2)

    x+yx+y, x,yS;

  3. (3)

    x=0 implies x=θ (the null vector of S), where is called the L 0 -norm on S and x is called the L 0 -norm of a vector xS.

It should be pointed out that the following idea of introducing the (ε,λ)-topology is due to Schweizer and Sklar [18].

Let (S,) be an RN module over K with base (Ω,E,P). For any positive real numbers ε and λ such that λ<1, let N θ (ε,λ)={xS|P{ωΩ|x(ω)<ε}>λ}, then { N θ (ε,λ)|ε>0,0<λ<1} is a local base at the null vector θ of some Hausdorff linear topology. The linear topology is called the (ε,λ)-topology. In this article, given an RN module (S,) over K with base (Ω,E,P), it is always assumed that (S,) is endowed with the (ε,λ)-topology. One only needs to notice that a sequence { x n ,nN} in S converges to xS in the (ε,λ)-topology if and only if { x n x,nN} converges to 0 in probability P.

Example Let X be a normed space over K and L 0 (E,X) the linear space of equivalence classes of X-valued E-random variables on Ω. The module multiplication operation : L 0 (E,K)× L 0 (E,X) L 0 (E,X) is defined by ξx= the equivalence class of ξ 0 x 0 , where ξ 0 and x 0 are the respective arbitrarily chosen representatives of ξ L 0 (E,K) and x L 0 (E,X), and ( ξ 0 x 0 )(ω)= ξ 0 (ω) x 0 (ω), ωΩ. Furthermore, the mapping : L 0 (E,X) L + 0 (E) by x=the equivalence class of x 0 , x L 0 (E,X), where x 0 is as above. Then it is easy to see that ( L 0 (E,X),) is an RN module over K with base (Ω,E,P).

Definition 2.2 [19]

Let ( S 1 , 1 ) and ( S 2 , 2 ) be two RN modules over K with base (Ω,E,P). A linear operator T from S 1 to S 2 is called a random linear operator, further, the random linear operator T is called a.s. bounded if there exists some ξ L + 0 (E) such that T x 2 ξ x 1 for any x S 1 . Denote by B( S 1 , S 2 ) the linear space of a.s. bounded random linear operators from S 1 to S 2 , define :B( S 1 , S 2 ) L + 0 (E) by T:={ξ L + 0 (E)| T x 2 ξ x 1 ,x S 1 } for any TB( S 1 , S 2 ), then it is easy to see that (B( S 1 , S 2 ),) is an RN module over K with base (Ω,E,P).

Specially, denote (B( S 1 , S 2 ),) by ( S , ) when ( S 1 , 1 ) is a given RN module (S,) over K with base (Ω,E,P) and S 2 = L 0 (E,K), then ( S , ) is called the random conjugate space of (S,). Let ( S , ) be the random conjugate space of ( S , ). The canonical embedding mapping J:S S defined by (Jx)(f)=f(x) for any xS and f S , is random-norm preserving. If J is surjective, then S is called random reflexive [10].

Definition 2.3 [8, 11]

Let (S,) be an RN module over K with base (Ω,E,P), B(S) the set of a.s. bounded random linear operators on S. A family {B(t):t0}B(S) is called a semigroup of random linear operators if

B(0)=IandB(s)B(t)=B(s+t)

for all s,t0, where I denotes the identity operator on S. Further, if the mapping B()x:[0,+)S, tB(t)x is continuous w.r.t. the (ε,λ)-topology for every xS, then the semigroup of random linear operators {B(t):t0} is said to be strongly continuous. Besides, if t 0 B(t) L + 0 (E), then {B(t):t0} is called an a.s. bounded strongly continuous semigroup of random linear operators.

Proposition 2.4 [19]

Let( S 1 , 1 )and( S 2 , 2 )be two RN modules over K with base(Ω,E,P). Then we have the following statements:

  1. (1)

    TB( S 1 , S 2 )if and only if T is a continuous module homomorphism;

  2. (2)

    IfTB( S 1 , S 2 ), thenT={ T x 2 :x S 1 and x 1 1}, where 1 denotes the identity element in L 0 (E).

3 Proof of the main results

The proof of Theorem 1.2 needs Theorem 1.1 and Lemma 3.5 below. To prove Theorem 1.1 and introduce Lemma 3.5, we will first recall the definition of Riemann integral for abstract-valued functions from a finite real interval to an RN module and a sufficient condition for such a function to be Riemann-integrable.

Let [a,b] be a finite real interval and P={ x 1 , x 2 ,, x k ,, x n } a finite partition into [a,b], namely, a= x 1 < x 2 << x n 1 < x n =b and λ(P)= max 1 i n {Δ x i }, where Δ x i = x i x i 1 (i=1,,n). Besides, in the following of this section we always suppose that (S,) denotes a complete RN module over K with base (Ω,E,P).

Definition 3.1 [8]

Let f be a function from [a,b] to S. f is called Riemann-integrable on [a,b] if there exists some I in S with the following property: for any positive numbers ε and λ with λ<1 there is a positive number δ(ε,λ) such that

P { ω Ω | i = 1 n f ( ξ i ) Δ x i I ( ω ) < ε } >1λ

for any finite partition P={ x 1 , x 2 ,, x k ,, x n } and arbitrarily chosen ξ i [ x i 1 , x i ] (1in) whenever λ(P)<δ(ε,λ). Further I is called the Riemann integral of f in the (ε,λ)-topology over [a,b], denoted by a b f(t)dt.

Proposition 3.2 [8]

Let f be a continuous function from[a,b]to S such that t [ a , b ] f(t) L + 0 (E), then f is Riemann integrable in the(ε,λ)-topology on[a,b]. Further, if t [ a , b ] f(t) L 2 (Ω,E,R), then f is Riemann integrable in 2 on[a,b], where 2 denotes the 2-norm of the Banach space L 2 (Ω,E,R).

Definition 3.3 [11]

Let {B(t):t0} be an a.s. bounded strongly continuous semigroup of random linear operators on an RN module S. We denote by

C(r)x:= 1 r 0 r B(s)xds,xS,r>0

the Cesàro means of {B(t):t0}. For any xS, if {C(r)x,r>0} converges to some point in S as r, then {B(t):t0} is called mean ergodic.

Note that it is Proposition 3.2 that makes Definition 3.3 be well defined.

Proposition 3.4 [11]

Let(S,)be a complete RN module over K with base(Ω,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 [9] that L 0 (E,X) is random reflexive if and only if X is a reflexive Banach space. Besides, Guo [3] proved that if an RN module S is random reflexive, then L F p (S) is also random reflexive. Based on these facts as well as the preceding Proposition 3.4, we can now prove Theorem 1.1 as follows.

Proof of Theorem 1.1 For each t0, define the mapping T(t): L F p (E,X) L F p (E,X) by T(t)x=x h t for each x L F p (E,X), then it is clear that T(t) is a module homomorphism. Furthermore,

T(0)x=x h 0 =x

for any x L F p (E,X) and

T(s)T(t)x=(x h t ) h s =x h t + s =T(s+t)x

for any s,t0 and x L F p (E,X), i.e. {T(t):t0} is a semigroup of random linear operators on L F p (E,X).

Since E( T ( t ) x p n|F),E( x p n|F) L 1 (Ω,F,P) for each t0, x L F p (E,X) and nN, it follows that

(3)

for any AF. Thus we have

E ( T ( t ) x p n | F ) =E ( x p n | F )
(4)

for any t0, x L F p (E,X) and nN, letting n in (4) yields that

E ( T ( t ) x p | F ) =E ( x p | F ) ,

namely T(t)x p p =x p p for any x L F p (E,X), which shows that

T(t): ( L F p ( E , X ) , p ) ( L F p ( E , X ) , p )

is a random isometric linear operator for each t0. Moreover,

lim t 0 x h t x p =0,x L F p (E,X).

Thus {T(t):t0} is a strongly continuous semigroup of random isometric linear operators on L F p (E,X).

Clearly, {T(t):t0} is an a.s. bounded strongly continuous semigroup of random linear operators on L F p (E,X). Since X is reflexive, it follows that the RN module L 0 (E,X) is random reflexive, further, L F p (E,X) is random reflexive. Thus it follows from Proposition 3.4 that T(t) is mean ergodic for each t0, i.e. for any x L F p (E,X) there exists some x ¯ L F p (E,X) such that

lim r 1 r 0 r (x h t )dt= lim r 1 r 0 r T(t)xdt= x ¯ .

Now it remains to show that x ¯ = x ¯ h t for each t0. In fact, since

T(s) ( 1 r 0 r T ( t ) x d t ) = 1 r 0 r T(s+t)xdt= 1 r s r + s T(t)xdt
(5)

and

(6)

for any r,s>0 and xS, fix s and x, letting r in (5) yields

lim r T(s) ( 1 r s r + s T ( t ) x d t ) = x ¯ .

Further, T(t) is a random isometric linear operator for each t0, thus we have T(t) x ¯ = x ¯ , namely x ¯ = x ¯ h t .

This completes the proof. □

It should be pointed out that Lemma 3.5 is merely mentioned in [8] and partially proved in [12].

Lemma 3.5 Let f be a continuous function from[a,b]to L 0 (E)such that t [ a , b ] |f(t)| L + 0 (E). Then

Ω [ a b f ( t ) d t ] dP= a b [ Ω f ( t ) d P ] dt.
(7)

Proof Let ξ= t [ a , b ] |f(t)|, then ξ L + 0 (E). Let H m =[m1ξ<m] for each mN, then { H m ,mN} is a sequence of pairwise disjoint F-measurable sets such that m = 1 H m =Ω. Define a mapping f m :[a,b]S by f m (t)= I H m f(t) for each mN. Obviously, t [ a , b ] | f m (t)| L 2 (Ω,E,R) for each mN and each t[a,b], thus it follows by Proposition 3.2 that f m is Riemann integrable in 2 on [a,b]. Let P, λ(P) and Δ x i , ξ i (1in) be the same as in Definition 3.1 and I m = a b f m (t)dt, I m n = i = 1 n f m ( ξ i )Δ x i for each mN. Since

(8)

for each fixed mN, we have

Ω I m dP= lim λ ( P ) 0 Ω I m n dP= lim λ ( P ) 0 i = 1 n Ω f m ( ξ i )dPΔ x i = a b [ Ω f m ( t ) d P ] dt.

Thus for any kN, we have

m = 1 k Ω [ a b f m ( t ) d t ] dP= m = 1 k a b [ Ω f m ( t ) d P ] dt.
(9)

Since

m = 1 P( H m )=P ( m = 1 H m ) =P(Ω),

it follows that { m = 1 k Ω [ a b f m (t)dt]dP,kN} and { m = 1 k a b [ Ω f m (t)dP]dt,kN} are both Cauchy sequences. Letting k in Equation (9) yields Equation (7). □

We can now prove Theorem 1.2.

Proof of Theorem 1.2 Let T(t)x=x h t for each t0 and x L F p ¯ (E,X), then it follows from Theorem 1.1 that

1 r 0 r (x h t )dt x ¯ (n)

in the (ε,λ)-topology induced by p ¯ . Thus by the random Schwarz-Cauchy inequality, we have

1 r 0 r (x h t )dt x ¯ 1 0(n)

in probability P. For any x L F 1 (E,X), since

1 r 0 r (x h t )dt 1 = 1 r 0 r T(t)xdt 1 x 1

for each nN and L F p ¯ (E,X) is dense in L F 1 (E,X) in 1 , one can obtain the desired Equation (2).

If x L 1 (Ω,E,X), it suffices to show that x ¯ =E(x|F). Let AF, then for each nN, it follows from Lemma 3.5 that

A [ 1 r 0 r T ( t ) x d t ] d P = 1 r 0 r [ A T ( t ) x d P ] d t = 1 r 0 r [ A x ( h t ( ω ) ) P ( d ω ) ] d t = 1 r 0 r [ h t 1 ( A ) x ( h t ( ω ) ) P ( d ω ) ] d t ( since h t 1 ( A ) = A ) = 1 r 0 r [ A x ( ω ) P ( d ω ) ] d t = A x ( ω ) P ( d ω ) .
(10)

Furthermore, since

lim n 1 r 0 r T(t)xdt= x ¯

in the (ε,λ)-topology induced by 1 and

1 r 0 r (x h t )dt 1 x 1

for each x L 1 (Ω,E,X) L F 1 (E,X), letting r in (10), 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. □

Let H be a Hilbert space over K. It is clear that Theorem 1.2 still holds when X is taken place of H, which includes the following known result.

Corollary 3.6 [12]

Let(Ω,E,P)be a probability space, { h t :t0}a semigroup of measure-preserving measurable isomorphisms on(Ω,E,P), F={AE| h t 1 (A)=A,t0}and lim t 0 x h t x 2 =0, x L F 2 (E,H). Then, for anyx L F 1 (E,H), there exists some x ¯ L F 1 (E,H)such that x ¯ = x ¯ h t for eacht0and

lim r 1 r 0 r (x h t )dt= x ¯ .
(11)

Corollary 3.7 If, in addition to the hypothesis of Theorem 1.2, we assume thatAFif and only ifP(A)=0of 1, then x ¯ =E(x).

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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.

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Zhang, X. On conditional mean ergodic semigroups of random linear operators. J Inequal Appl 2012, 150 (2012). https://doi.org/10.1186/1029-242X-2012-150

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Keywords

  • Equivalence Class
  • Ergodic Theorem
  • Continuous Semigroup
  • Reflexive Banach Space
  • Linear Topology