On conditional mean ergodic semigroups of random linear operators
© Zhang; licensee Springer 2012
Received: 7 February 2012
Accepted: 8 May 2012
Published: 3 July 2012
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.
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.
in the-topology induced by.
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.
, and ;
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 .
Definition 2.2 
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 .
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.
Proposition 2.4 
if and only if T is a continuous module homomorphism;
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 .
Definition 3.1 
for any finite partition and arbitrarily chosen () whenever . Further I is called the Riemann integral of f in the -topology over , denoted by .
Proposition 3.2 
Let f be a continuous function fromto S such that, then f is Riemann integrable in the-topology on. Further, if, then f is Riemann integrable inon, wheredenotes the 2-norm of the Banach space.
Definition 3.3 
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.
Proposition 3.4 
Letbe a complete RN module over K with base. 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  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.
for any and , i.e. is a semigroup of random linear operators on .
Thus is a strongly continuous semigroup of random isometric linear operators on .
Further, is a random isometric linear operator for each , thus we have , namely .
This completes the proof. □
We can now prove Theorem 1.2.
for each and is dense in in , one can obtain the desired Equation (2).
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 
Corollary 3.7 If, in addition to the hypothesis of Theorem 1.2, we assume thatif and only ifof 1, then.
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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