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Mean ergodic theorem for semigroups of linear operators in multi-Banach spaces
Journal of Inequalities and Applications volume 2014, Article number: 402 (2014)
In this paper, by using Rodé’s method, we extend Yosida’s theorem to semigroups of linear operators in multi-Banach spaces.
MSC:39A10, 39B72, 47H10, 46B03.
In 1938, Yosida  proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and () be linear operators of E into itself such that there exists a constant C with for , and is weakly completely continuous, i.e., maps the closed unite ball of E into a weakly compact subset of E. Then the Cesaro means
converges strongly as to a fixed point of for each .
On the other hand, in 1975, Baillon  proved the following nonlinear ergodic theorem. Let X be a Banach space and C be a closed convex subset of X. The mappings () are called nonexpansive on C if
Let be the set of fixed points of . If X is strictly convex, is closed and convex. In , Baillon proved the first nonlinear ergodic theorem such that if X is a real Hilbert space and , then for each , the sequence defined by
converges weakly to a fixed point of . It was also shown by Pazy  that if X is a real Hilbert space and converges weakly to , then .
Recently, Rodé  and Takahashi  tried to extend this nonlinear ergodic theorem to semigroup, generalizing the Cesaro means on , such that the corresponding sequence of mappings converges to a projection onto the set of common fixed points. In this paper, by using Rodé’s method, we extend Yosida’s theorem to semigroups of linear operators in multi-Banach spaces. The proofs employ the methods of Yosida , Rodé , Greenleaf  and Takahashi [7, 8]. Our paper is motivated from ideas in .
2 Multi-Banach spaces
The notion of multi-normed space was introduced by Dales and Polyakov in . This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [10–16].
Let be a complex normed space, and let . We denote by the linear space consisting of k-tuples , where . The linear operations on are defined coordinate-wise. The zero element of either E or is denoted by 0. We denote by the set and by the group of permutations on k symbols.
Definition 2.1 Let E be a linear space, and take . For , define
For , define
Definition 2.2 Let be complex (respectively, real) normed space, and take . A multi-norm of level n on is a sequence such that is a norm on for each , such that for each (so that is the initial norm), and such that the following axioms (A1)-(A4) are satisfied for each with :
(A1) for each and , we have
(A2) for each (respectively, each ) and , we have
(A3) for each , we have
(A4) for each
In this case, is a multi-normed space of level n.
A multi-norm on is a sequence
such that is a multi-norm of level n for each . In this case, is a multi-normed space.
Lemma 2.3 
Suppose that is a multi-normed space, and take . Then
It follows from (b) that, if is a Banach space, then is a Banach space for each ; in this case is a multi-Banach space.
Now we state two important examples of multi-norms for an arbitrary normed space E; cf. .
Example 2.4 The sequence on defined by
is a multi-norm called the minimum multi-norm. The terminology ‘minimum’ is justified by property (b).
Example 2.5 Let be the (non-empty) family of all multi-norms on . For , set
Then is a multi-norm on , called the maximum multi-norm.
We need the following observation, which can easily be deduced from the triangle inequality for the norm and the property (b) of multi-norms.
Lemma 2.6 Suppose that and . For each , let be a sequence in E such that . Then for each we have
Definition 2.7 Let be a multi-normed space. A sequence in E is a multi-null sequence if, for each , there exists such that
Let . We say that the sequence is multi-convergent to and write
if is a multi-null sequence.
3 Preliminaries and lemmas
Let E a real Banach space and let be the conjugate space of E, that is, the space of all continuous linear functionals on E. The value of at will be denoted by . We denote by coD the convex hull of D, the closure of coD.
Let U be a linear continuous operator of E into itself. Then we denote by the conjugate operator of U.
Assumption (A) Let be a multi-Banach space and () be a family of linear continuous operators of a real Banach space into itself such that there exists a real number C with for all and the weak closure of is weakly compact, for each . The index set G is a topological semigroup such that for all and is continuous with respect to the weak operator topology: for all and if in G.
We denote by the Banach space of all bounded continuous real valued functions on the topological semigroup G with the supremum norm. For each and , we define elements and in given by and for all . An element (the conjugate space of ) is called a mean on G if moreover, we have . A mean on G is called left (right) invariant if () for all and . An invariant mean is a left and right invariant mean. We know that is a mean on G if and only if
Let be a family of linear continuous operators of E into itself satisfying Assumption (A) and be a mean on G. Fix . Then, for , the real valued function is in . Denote by the value of at this function. By linearity of and of , this is linear in ; moreover, since
it is continuous in . Hence we find that is an element of . So, from weak compactness of such that for every .
Put and suppose that the element is not contained in the , where n is the natural embedding of the Banach space E into its second conjugate space . Then, since the convex set is compact in the weak∗ topology of , there exists an element such that
Hence, we have
This is a contradiction. Thus, for a mean on G, we can define a linear continuous operator of E into itself such that , for all , and for all and . We denote by the set all common fixed points of the mappings , .
Lemma 3.1 Assume that a left invariant mean exists on G, then . Especially, is then not empty.
Proof Let and μ be a left invariant mean on G. Then since, for and ,
we have . Hence, . □
Lemma 3.2 Let be an invariant mean on G. Then for each and for each mean on G. Especially, is a projection of E onto .
Proof Let . Then, since
for and , we have . It is obvious from Lemma 3.1 that for each . Let be a mean on G. Then, since
for and , we have . Putting , we have and hence is a projection of E onto . □
As direct consequence of Lemma 3.2, we have the following.
Lemma 3.3 Let and be invariant means on G. Then .
Lemma 3.4 Assume that an invariant mean exists on G. Then, for each , the set consists of a single point.
Proof Let and be an invariant mean on G. Then we know that and . So, we show that . Let and . Then, for , there exists an element in the set such that . Hence, we have
Since ϵ is arbitrary, we have for every and hence . □
4 Ergodic theorems
Now, we can prove mean ergodic theorems for semigroups of linear continuous operators in multi-Banach space.
Theorem 4.1 Let be a family of linear continuous operators in a real Banach space E satisfying Assumption (A). If a net of means on G is asymptotically invariant, i.e.,
converge to 0 in the weak∗ topology of for each , then there exists a projection of E on to such that , converges weakly to for each , for each , and for each . Furthermore, the projection onto is the same for all asymptotically invariant nets.
Proof Let be a cluster point of net in the weak∗ topology of . Then is an invariant mean on G. Hence, by Lemma 3.2, is a projection of E onto such that , for each and for each . Setting , we show that converges weakly to for each . Since for all and is weakly compact, there exists a subnet of such that converges weakly to an element . To show that converges weakly to , it is sufficient to show . Let and . Since weakly, we have and . On the other hand, since in the weak∗ topology, we have
Hence, we have and hence . So, we obtain from Lemma 3.4. That the projection is the same for all asymptotically invariant nets is obvious from Lemma 3.3. □
As direct consequence of Theorem 4.1, we have the following.
Corollary 4.2 Let be as in Theorem 4.1 and assume that an invariant mean exists on G. Then there exists a projection of E onto such that , for each and for each .
Theorem 4.3 Let be as in Theorem 4.1. If a net of means on G is asymptotically invariant and further converges to 0 in the strong topology of , then there exists a projection of E onto such that , converges strongly to for each , for each , and for each .
Proof As in the proof of Theorem 4.1, let , where is a cluster point of the net in the weak∗ topology of . We show that converges strongly to for each .
Let . Then, for any , converges strongly to 0. In fact, if , then since, for any ,
we have . Using this inequality, we show that converges strongly to 0 for any . Let z be any element of and ϵ be any positive number. By the definition of , there exists an element in the set such that . On the other hand, from for all , there exists such that, for all and ,
Hence, converges strongly to 0 for each .
Next, assume that for some is not contained in the set . Then, by the Hahn-Banach theorem, there exists a linear continuous functional such that and for all . So since for all , we have
This is a contradiction. Hence, for all are contained in . Therefore we find that converges strongly to 0 for all . This completes the proof. □
By using Theorem 4.3, we can obtain the following corollary.
Corollary 4.4 Let E be a real Banach space and be a linear operator of E into itself such that exists a constant C with for , and is weakly completely continuous, i.e., maps the closed unit ball of E into a weakly compact subset of E. Then there exists a projection of E onto the set of all fixed point of such that , the Cesaro means converges strongly to for each , and .
Proof Let . Then, since , where means the closed ball with center x and radius r, the weak closure of is weakly compact. On the other hand, let with the discrete topology and be a mean on G such that for each . Then it is obvious that for all . So, it follows from Theorem 4.3 that Corollary 4.4 is true. □
If with the natural topology, then we obtain the corresponding result.
Corollary 4.5 Let E be a real Banach space and be a family of linear operators of E into itself satisfying Assumption (A). Then there exists a projection of E onto such that , converges strongly to for each , and for each .
Remark 4.6 are weak vector valued integrals with respect to means on . As in Section IV of Rodé , we can also obtain the strong convergence of the sequences
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Yeol Je Cho was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2013053358).
The authors declare that they have no competing interests.
All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Kenari, H.M., Saadati, R., Azhini, M. et al. Mean ergodic theorem for semigroups of linear operators in multi-Banach spaces. J Inequal Appl 2014, 402 (2014). https://doi.org/10.1186/1029-242X-2014-402
- ergodic theorem
- multi-Banach space