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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005
A Note on the L1-Mean Ergodic Theorem
María Elena Becker
Universidad de Buenos Aires, Argentina.
Abstract:
Let be a positive contraction on of a -finite measure space. Necessary and sufficient conditions are given in order that for any in the averages converge in the norm of .
1. Introduction.
Let be a -finite measure space and let denote the usual Banach space of all real-valued integrable functions on . A linear operator is called positive if implies , and a contraction if , with denoting the operator norm of on . We say that the pointwise ergodic theorem (resp. the -mean ergodic theorem) holds for if for any in the ergodic averages
converge a.e. on (resp. in -norm).
In 1964, Chacon [1] showed a class of positive contractions on for which the pointwise ergodic theorem fails to hold and Ito [4] proved that if a positive contraction on satisfies the -mean ergodic theorem, then it satisfies the pointwise ergodic theorem. (cf. also Kim [5]).
More recently, Hasegawa and Sato [2] proved that if are commuting positive contractions on such that each , , satisfies the -mean ergodic theorem, and if are (not necessarily commuting) contractions on such that , for , then the averages
converge a.e. for every in .
In view of these facts, it seems interesting to find out conditions on which guarantee the validity of the -mean ergodic theorem.
In order to state our result, we will need some definitions and previous results. Let us fix some notation. We denote the space of all nonnegative functions in and the space of all nonnegative functions in by and , respectively. The adjoint operator of , which acts on , is denoted by . Put
By Hopf decomposition, , where and denote respectively the conservative and the dissipative part of respect to . We recall that and are determined uniquely mod by:
C1) For all , a.e. on , and
D1) For all , a.e on .
For any , will be the characteristic function of the set . Let us write
From the results of Helmberg [3] and Lin and Sine [7] about the relationship between the validity of the -mean ergodic theorem for and the almost everywhere convergence of the averages , , we have:
Theorem 1.1. [Helmberg; Lin-Sine] Let be a positive contraction on . Then the following are equivalent:- The -mean ergodic theorem holds for .
- For any in , the averages converge a.e.
- a.e. and there exists a nonnegative function in satisfying and .
Kim [5] has proved:
Theorem 1.2. [Kim] Let be a positive contraction on . Suppose that the sequence is weakly sequentially compact for some in . Then for each in , exists in the -norm and almost everywhere.
The purpose of this paper is to prove the following result:
Theorem A. Let be a positive contraction on . Then the following assertions are equivalent:
- The -mean ergodic theorem holds for .
- a.e. and there exists in such that and the sequence is weakly sequentially compact.
- There exists in such that the averages converge a.e. to a function and a.e.
We refer the reader to Krengel's book [6] for a proof of the following properties related with Hopf decomposition of :
P1) For all , a.e. on .
P2) If and , then a.e. on .
P3) a.e.
By P) and P) we see that a.e. on .
We start with the following lemmas.
Lemma 2.1. Let be a function in such that the averages converge a.e. Let us denote the limit function by . Then a.e.
Proof. Because of the identity
By D) we have a.e. on and therefore on . The lemma is proved.
Lemma 2.2. Let in such that and let , . Then we have:
i) a.e.
ii) and , with such that and a.e.
Proof. Being a positive contraction on , i) follows from
We are now ready to prove our result.
Proof of Theorem A. By virtue of theorem 1.1 it is sufficient to prove that the following assertions are equivalent:
i) a.e. and there exists in such that and the sequence is weakly sequentially compact.
ii) There exists in such that the averages converge a.e. to a function and a.e.
iii) a.e. and there exists in satisfying and .
The implications iii) ii) and iii) i) are immediate.
i) iii) By the mean ergodic theorem of Yosida and Kakutani [8], the sequence converges in -norm to a function such that . Put and . By i) of lemma 2.2 there exists the a.e. and . Since (see e.g. [3]) we have:
where the third equality follows from the fact that being -norm convergent is weakly convergent in .
Then . Moreover, from a.e., we can see that a.e.
On the other hand, from ii) of lemma 2.2 we obtain on , for all . As on , we deduce that for all on . Therefore, for all , a.e. on . Thus and iii) follows.
ii) iii) By lemma 2.1, . Put and . By ii) of lemma 2.2, . Then a.e. and a.e. Now, iii) follows as in i) iii). □
Remarks.- In iii) of Theorem A, the condition , can not be replaced by . To see this, take an ergodic, conservative measure preserving transformation with respect to , where is -finite and infinite. Then, the operator satisfies the pointwise ergodic theorem, but the -mean ergodic theorem does not hold for .
- Suppose the -mean ergodic theorem holds for . Then a.e. For each in , we denote by the a.e. limit of . Now, let in . Since , we have
Proposition 2.3. The following assertions are equivalent:
i) a.e.
ii) Let in . Then if and only if and on .
Sketch of proof.
i) ii) follows from a.e. and the fact that implies .
ii) i) let . Then and .
References
[1] R. Chacon, A class of linear transformations, Proc. Amer. Math. Soc. 15 (1964), 560-564. [ Links ]
[2] S. Hasegawa and R. Sato, On d-parameter pointwise ergodic theorems in L1, Proc. Amer. Math. Soc. 123(1995), 3455-3465. [ Links ]
[3] G. Helmberg, On the Converse of Hopf's Ergodic Theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 (1972), 77-80. [ Links ]
[4] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222-227. [ Links ]
[5] C. Kim, A generalization of Ito's theorem concerning the pointwise ergodic theorem, Ann. Math. Statist. 39 (1968), 2145-2148. [ Links ]
[6] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. [ Links ]
[7] M. Lin and R. Sine, The individual ergodic theorem for non-invariant measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), 329-331. [ Links ]
[8] K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Ann. Math 42 (1941), 188-228. [ Links ]
María Elena Becker
Departamento de Matemática
Facultad de Ciencias Exactas y Naturales
UNIVERSIDAD DE BUENOS AIRES
Pab. I, Ciudad Universitaria
(1428) Buenos Aires, Argentina.
mbecker@dm.uba.ar
Recibido: 14 de noviembre de 2004
Aceptado: 10 de noviembre de 2005