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  and subsequently elaborated in , 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 [3-9], 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 [6,8,10-12].
It is well known that the -topology induced by the -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 -topology, studied probability theory in Banach spaces and did many excellent works [13,14] under the framework of the special RN module , where is the RN module of equivalence classes of X-valued random variables defined on a probability space , see  or Section 2 for the construction of . 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  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 . 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 and the construction of is detailed as follows.
Now let us first recall the construction of in . Let be a probability space, a sub σ-algebra of and (or ) the set of equivalence classes of -measurable extended real-valued (real-valued) random variables on Ω. Let and . Similarly, one can understand such notions as and . Define the mapping by
for any and , where denotes the extended conditional expectation, and let
then is an RN module. In fact, is exactly the -module generated by , namely , where . Further, motivated by the idea of constructing , Guo in  constructed a more general RN module and established the random conjugate representation theorem of this type of RN module. Precisely speaking, let be an RN module over the scalar field K with base , the mapping is defined as follows for any and :
where also denotes the extended conditional expectation and let
Then is an RN module over K with base . If we take , then is exactly ; if we further take , then (briefly, ) is an RN module over K with base .
We can now state the main results of this article as follows.
Theorem 1.1Let be a probability space, Xa reflexive Banach space overK, a semigroup of measure-preserving measurable isomorphisms on , pan arbitrary positive number such that and . If , , then, for any , there exists some such that for each and
in the -topology induced by .
Theorem 1.2Let , X, and be the same as in Theorem 1.1 and a given positive number such that and , . Then, for any , there exists some such that for each and
in the -topology induced by . In particular, if , then .
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.
In the sequel of this article, denotes a probability space, K the scalar field R of real numbers or C of complex numbers, and the algebra of equivalence classes of K-valued -measurable random variables on Ω under the ordinary addition, scalar multiplication and multiplication operations on equivalence classes. Besides, let , and be the same as in Section 1.
It is well known from  that is a complete lattice under the ordering ≤: if and only if for P-almost all ω in Ω (briefly, a.s.), where and are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset A of has a supremum, denoted by ⋁A, and an infimum, denoted by ⋀A, and there exist two sequences and in A such that and . Finally, , as a sublattice of , is complete in the sense that every subset with an upper bound has a supremum.
An ordered pair is called an RN module over K with base if S is a left module over the algebra and is a mapping from S to such that the following three axioms are satisfied:
(1) , and ;
(2) , ;
(3) implies (the null vector of S), where is called the -norm on S and is called the -norm of a vector .
It should be pointed out that the following idea of introducing the -topology is due to Schweizer and Sklar .
Let be an RN module over K with base . For any positive real numbers ε and λ such that , let , then is a local base at the null vector θ of some Hausdorff linear topology. The linear topology is called the -topology. In this article, given an RN module over K with base , it is always assumed that is endowed with the -topology. One only needs to notice that a sequence in S converges to in the -topology if and only if converges to 0 in probability P.
Example Let X be a normed space over K and the linear space of equivalence classes of X-valued -random variables on Ω. The module multiplication operation is defined by the equivalence class of , where and are the respective arbitrarily chosen representatives of and , and , . Furthermore, the mapping by , , where is as above. Then it is easy to see that is an RN module over K with base .
Let and be two RN modules over K with base . A linear operator T from to is called a random linear operator, further, the random linear operator T is called a.s. bounded if there exists some such that for any . Denote by the linear space of a.s. bounded random linear operators from to , define by for any , then it is easy to see that is an RN module over K with base .
Specially, denote by when is a given RN module over K with base and , then is called the random conjugate space of . Let be the random conjugate space of . The canonical embedding mapping defined by for any and , is random-norm preserving. If J is surjective, then S is called random reflexive .
Let be an RN module over K with base , the set of a.s. bounded random linear operators on S. A family is called a semigroup of random linear operators if
for all , where I denotes the identity operator on S. Further, if the mapping , is continuous w.r.t. the -topology for every , then the semigroup of random linear operators is said to be strongly continuous. Besides, if , then is called an a.s. bounded strongly continuous semigroup of random linear operators.
Let and be twoRNmodules overKwith base . Then we have the following statements:
(1) if and only ifTis a continuous module homomorphism;
(2) If , then , where 1 denotes the identity element in .
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 be a finite real interval and a finite partition into , namely, and , where ( ). Besides, in the following of this section we always suppose that denotes a complete RN module over K with base .
Let f be a function from to S. f is called Riemann-integrable on if there exists some I in S with the following property: for any positive numbers ε and λ with there is a positive number such that
for any finite partition and arbitrarily chosen ( ) whenever . Further I is called the Riemann integral of f in the -topology over , denoted by .
Letfbe a continuous function from toSsuch that , thenfis Riemann integrable in the -topology on . Further, if , thenfis Riemann integrable in on , where denotes the 2-norm of the Banach space .
Let be an a.s. bounded strongly continuous semigroup of random linear operators on an RN module S. We denote by
the Cesàro means of . For any , if converges to some point in S as , then is called mean ergodic.
Note that it is Proposition 3.2 that makes Definition 3.3 be well defined.
Let be a completeRNmodule overKwith base . IfSis random reflexive, then every a.s. bounded strongly continuous semigroup of random linear operators onSis mean ergodic.
It is known from  that is random reflexive if and only if X is a reflexive Banach space. Besides, Guo  proved that if an RN module S is random reflexive, then 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 , define the mapping by for each , then it is clear that is a module homomorphism. Furthermore,
for any and
for any and , i.e. is a semigroup of random linear operators on .
Since for each , and , it follows that
for any . Thus we have
for any , and , letting in (4) yields that
namely for any , which shows that
is a random isometric linear operator for each . Moreover,
Thus is a strongly continuous semigroup of random isometric linear operators on .
Clearly, is an a.s. bounded strongly continuous semigroup of random linear operators on . Since X is reflexive, it follows that the RN module is random reflexive, further, is random reflexive. Thus it follows from Proposition 3.4 that is mean ergodic for each , i.e. for any there exists some such that
Now it remains to show that for each . In fact, since
for any and , fix s and x, letting in (5) yields
Further, is a random isometric linear operator for each , thus we have , namely .
This completes the proof. □
Lemma 3.5Letfbe a continuous function from to such that . Then
Proof Let , then . Let for each , then is a sequence of pairwise disjoint -measurable sets such that . Define a mapping by for each . Obviously, for each and each , thus it follows by Proposition 3.2 that is Riemann integrable in on . Let , and , ( ) be the same as in Definition 3.1 and , for each . Since
for each fixed , we have
Thus for any , we have
it follows that and are both Cauchy sequences. Letting in Equation (9) yields Equation (7). □
We can now prove Theorem 1.2.
Proof of Theorem 1.2 Let for each and , then it follows from Theorem 1.1 that
in the -topology induced by . Thus by the random Schwarz-Cauchy inequality, we have
in probability P. For any , since
for each and is dense in in , one can obtain the desired Equation (2).
If , it suffices to show that . Let , then for each , it follows from Lemma 3.5 that
in the -topology induced by and
for each , letting in (10), it follows from the Lebesgue’s dominated convergence theorem that
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.
Let be a probability space, a semigroup of measure-preserving measurable isomorphisms on , and , . Then, for any , there exists some such that for each and
Corollary 3.7If, in addition to the hypothesis of Theorem 1.2, we assume that if and only if of 1, then .
The author declare that they have no competing interests.
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.
Guo, TX: Extension theorems of continuous random linear operators on random domains. J. Math. Anal. Appl.. 193, 15–27 (1995). Publisher Full Text
Guo, TX: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal.. 258, 3024–3047 (2010). Publisher Full Text
Guo, TX: Recent progress in random metric theory and its applications to conditional risk measures. Sci. China Ser. A. 54(4), 633–660 (2011). Publisher Full Text
Guo, TX, Zeng, XL: Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal.. 73, 1239–1263 (2010). Publisher Full Text
Guo, TX, Li, SB: The James theorem in complete random normed modules. J. Math. Anal. Appl.. 308, 257–265 (2005). Publisher Full Text
Guo, TX, Shi, G: The algebraic structure of finitely generated -modules and the Helly theorem in random normed modules. J. Math. Anal. Appl.. 381, 833–842 (2011). Publisher Full Text
in ChinesePublisher Full Text
Zhang, X: On mean ergodic semigroups of random linear operators. Proc. Jpn. Acad., Ser. A, Math. Sci.. 88, 53–58 (2012). Publisher Full Text
Zhang, X, Guo, TX: Von Neumann’s mean ergodic theorem on complete random inner product modules. Front. Math. China. 6(5), 965–985 (2011). Publisher Full Text
Taylor, RL: Convergence of elements in random normed spaces. Bull. Aust. Math. Soc.. 12, 31–47 (1975). Publisher Full Text
Beck, A, Schwartz, JT: A vector-valued random ergodic theorem. Proc. Am. Math. Soc.. 8, 1049–1059 (1957). Publisher Full Text
Filipović, D, Kupper, M, Vogelpoth, N: Separation and duality in locally -convex modules. J. Funct. Anal.. 256, 3996–4029 (2009). Publisher Full Text