𝐿⁰-Linear Modulus of a Random Linear Operator

Abstract

We prove that there exists a unique -linear modulus for an a.s. bounded random linear operator on a specifical random normed module, which generalizes the classical case.

1. Introduction

In 1964, Chacon and Krengel began to study linear modulus of a linear operator and proved that there exists a unique linear modulus for a bounded linear operator [1], which plays an important role in the work of mean ergodicity for linear operators and linear operators semigroups [2–5]. Recently, the mean ergodicity for random linear operators has been investigated in [6–8], and its further developments should naturally include the study of -linear modulus of a random linear operator on a random normed module. The purpose of this paper is to investigate the existence of the -linear modulus for an a.s. bounded random linear operator on a specifical random normed module.

The notion of random normed modules (briefly, RN modules), which was first introduced in [9] and subsequently elaborated in [10], is a random generalization of ordinary normed spaces. In the last ten years the theory of RN modules together with their random conjugate spaces has obtained systematic and deep developments [11–17]; in particular, the recently developed -convex analysis, which has been a powerful tool for the study of conditional risk measures, is just based on the theory of RN modules together with their random conjugate spaces [12, 17–20]. One of the key points in the process of applying the theory of RN modules to random analysis and the theory of conditional risk measures is to properly construct the two classes of RN modules and , where is the RN module of equivalence classes of -valued random variables defined on a probability space and is the -module generated by ; see [17], for the construction of . In particular, constructed in [18] will be used in this paper and thus we give the details of its construction as follows.

Let be a probability space, a sub -algebra of , and (or ) the set of equivalence classes of -measurable extended real-valued (real-valued) random variables on . Let and . Similarly, one can understand such notations 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 .

The remainder of this paper is organized as follows: in Section 2 we briefly recall some necessary notions and facts and in Section 3 we present and prove our main results.

2. Preliminaries

In the sequel of this paper, denotes a given probability space, the scalar field of real numbers or of complex numbers, the set of positive integers, and the algebra over of equivalence classes of -valued -measurable random variables on under the ordinary scalar multiplication, addition, and multiplication operations on equivalence classes.

Proposition 1 (see [21]). , is a complete lattice under the ordering if and only if , for -almost all in (briefly, a.s.), where and are arbitrarily chosen representatives of and , respectively, and has the following nice properties.
(1) Every subset of has a supremum (denoted by ) and an infimum (denoted by ) and there exist two sequences and in such that and .
(2) If is directed (dually directed), namely, for any two elements and in , there exists some in such that (); then the above () can be chosen as nondecreasing (nonincreasing).
(3) , as a sublattice of , is complete in the sense that every subset with an upper bound (a lower bound) has a supremum (an infimum).

Let and be two elements in ; then is understood as usual, namely, and . For , on means   -a.s. on , where and are arbitrarily chosen representatives of and , respectively. Specially, we denote and on }.

Definition 2 (see [10, 17]). An ordered pair is called a random normed module (briefly, an RN module) over with base if is a left module over the algebra and is a mapping from to such that the following three axioms are satisfied:(1) if and only if (the null vector of );(2), for all and ;(3), for all .

Cleary, is an RN module over with base .

Let be an RN module over with base . For any , , denote ; then is a local base at of some Hausdorff linear topology, called the -topology induced by . In this paper, given an RN module over with base , it is always assumed that is endowed with the -topology. In this paper, it suffices to notice that the -topology for an RN module is a metrizable linear topology and a sequence in converges in the -topology to some if and only if converges in probability to 0. It should be pointed out that the -topology for is exactly the topology of convergence in probability.

Example 3. Let be a normed space over and the linear space of equivalence classes of -valued -random variables on . The module multiplication operation is defined by the equivalence class of , where and are the respective arbitrarily chosen representatives of and , and , for all . Furthermore, the mapping by = the equivalence class of , for all , where is as above. Then it is easy to see that is an RN module over with base .

Definition 4 (see [22]). Let and be two RN modules over with base . A linear operator from to is called a random linear operator; further, the random linear operator 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 all for any ; then it is easy to see that is an RN module over with base .

Proposition 5 (see [22]). Let and be two RN modules over with base . Then, we have the following statements:(1)  if and only if is a continuous module homomorphism;(2)if  , then    and  , where denotes the identity element in  .

3. Main Results and Proofs

The main result of this paper is Theorem 7, which will be derived from Lemma 6.

Let be the class of simple -measurable functions and ; then Lemma 6 holds.

Lemma 6. is dense in .

Proof. For any , there exist and such that Clearly, ; thus there exists a sequence such that as ; that is, . on as . Since , it follows that Furthermore, since , it follows that converges to 0 a.s. on as . Hence, converges to 0 in probability as . Consequently, converges to 0 in the -topology as .
Next, observe that ; it follows from the above discussion that there exists a sequence such that converges to in the -topology induced by as .
Let Then, and converges to in the -topology induced by as , which shows that is dense in .

Now we can present and prove the main result below.

Theorem 7. Let be an a.s. bounded random linear operator on . Then there exists a unique positive a.s. bounded random linear operator on , called the -linear modulus of , such that(1),(2) for any ,(3) for any .

Proof. Let denote the family of all finite measurable partitions of to ; that is, for any , there exists   () such that , where is a finite number with respect to . It is known that is partially ordered in the usual way: in means that is a refinement of ; that is, the sets are unions of sets of . For any , define Then, for any fixed , is monotone increasing on . Furthermore, letting in (9), yields that Observe that Combining (10) and (11), we have which shows that the net is not only monotone increasing on but also -norm bounded with respect to . Thus, we can define by Then, is a positive a.s bounded random linear operator according to Lemma 6 and from inequality (12) we get .
For any , let For any and , it is clear that , and further observe that since is positive. Consequently, which shows that holds. If the converse inequality of (16) does not hold, then there exists an , a , an with , and an such that
Now there exists a set with and a with such that on . Continuing in this way we find with and with () such that on . Setting we have and this leads to a contradiction with inequality (17) on .
This completes the proof.

If we put , then the following classical result holds.

Corollary 8 (see [1]). Let be a bounded linear operator on . Then, there exists a unique positive bounded linear operator on , called the linear modulus of , such that(1),(2) for any ,(3)  and   for any .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere gratitude to Professor Guo Tiexin for his invaluable suggestions. This research is supported by the National Natural Science Foundation of China (Grant no. 11301380) and the Higher School Science and Technology Development Fund Project in Tianjin (Grant no. 20031003).