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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
![1 n∑-1 An(T )f = -- Tif n i=0](/img/revistas/ruma/v46n1/1a0530x.png)
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
![An(T1, ...,Td)f = An(T1) ...An(Td)f](/img/revistas/ruma/v46n1/1a0546x.png)
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
![∞∑ ∑∞ S ∞f = T if, f ∈ L+1 ; S*∞h = T *ih, h ∈ L+∞. i=0 i=0](/img/revistas/ruma/v46n1/1a0558x.png)
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
![n-1 * 1-∑ *i An(T )h = n T h, h ∈ L∞. i=0](/img/revistas/ruma/v46n1/1a0578x.png)
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:
![T](/img/revistas/ruma/v46n1/1a0583x.png)
![L1](/img/revistas/ruma/v46n1/1a0584x.png)
- 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
.
![K](/img/revistas/ruma/v46n1/1a0595x.png)
![L1](/img/revistas/ruma/v46n1/1a0596x.png)
![{φn}](/img/revistas/ruma/v46n1/1a0597x.png)
![K](/img/revistas/ruma/v46n1/1a0598x.png)
![{φnk}](/img/revistas/ruma/v46n1/1a0599x.png)
![L1](/img/revistas/ruma/v46n1/1a05100x.png)
![φ](/img/revistas/ruma/v46n1/1a05101x.png)
![L1](/img/revistas/ruma/v46n1/1a05102x.png)
![h](/img/revistas/ruma/v46n1/1a05103x.png)
![L ∞](/img/revistas/ruma/v46n1/1a05104x.png)
![∫ ∫ lim hφ dμ = hφ dμ. k nk](/img/revistas/ruma/v46n1/1a05105x.png)
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
![n-+-1- f- T An(T )f = n An+1(T )f - n ,](/img/revistas/ruma/v46n1/1a05150x.png)
![lim T An(T )f = f0](/img/revistas/ruma/v46n1/1a05151x.png)
![f0 ∈ L1](/img/revistas/ruma/v46n1/1a05152x.png)
![An(T )f ∧ f0 := min{An(T )f,f0}](/img/revistas/ruma/v46n1/1a05153x.png)
![f0](/img/revistas/ruma/v46n1/1a05154x.png)
![L1](/img/revistas/ruma/v46n1/1a05155x.png)
![T (An(T )f ∧ f0)](/img/revistas/ruma/v46n1/1a05156x.png)
![Tf0](/img/revistas/ruma/v46n1/1a05157x.png)
![L1](/img/revistas/ruma/v46n1/1a05158x.png)
![T f0 ≤ f0](/img/revistas/ruma/v46n1/1a05159x.png)
![S∞(f0 - Tf0) ≤ f0](/img/revistas/ruma/v46n1/1a05160x.png)
![1](/img/revistas/ruma/v46n1/1a05161x.png)
![f0 = T f0](/img/revistas/ruma/v46n1/1a05162x.png)
![C](/img/revistas/ruma/v46n1/1a05163x.png)
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
![∫ ∫ * 0 = χD0f dμ = fT χD0 dμ.](/img/revistas/ruma/v46n1/1a05182x.png)
![1)](/img/revistas/ruma/v46n1/1a05183x.png)
![C0 ⊂ C](/img/revistas/ruma/v46n1/1a05184x.png)
![∫ ∫ ∫ * f dμ = f χC0 dμ = fT χC0 d μ,](/img/revistas/ruma/v46n1/1a05185x.png)
![* T χC0 = 1](/img/revistas/ruma/v46n1/1a05186x.png)
![C0](/img/revistas/ruma/v46n1/1a05187x.png)
![* T χC0 = χC0 + h](/img/revistas/ruma/v46n1/1a05188x.png)
![h](/img/revistas/ruma/v46n1/1a05189x.png)
![+ L ∞](/img/revistas/ruma/v46n1/1a05190x.png)
![{h > 0} ⊂ D0](/img/revistas/ruma/v46n1/1a05191x.png)
![n](/img/revistas/ruma/v46n1/1a05192x.png)
![n∑-1 T *nχ = χ + T *kh. C0 C0 k=0](/img/revistas/ruma/v46n1/1a05193x.png)
![* *n S∞h = lim T χC0 - χC0 ≤ 1](/img/revistas/ruma/v46n1/1a05194x.png)
![1](/img/revistas/ruma/v46n1/1a05195x.png)
![{h > 0} ⊂ D](/img/revistas/ruma/v46n1/1a05196x.png)
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:
![∫ ∫ ∫ ∫ 0 = uw0 dμ = ulim An(T )w dμ = lim uAn(T )w dμ = uw d μ](/img/revistas/ruma/v46n1/1a05224x.png)
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). □
- 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
. Then, for all
a.e.
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