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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 T  be a positive contraction on L1   of a σ  -finite measure space. Necessary and sufficient conditions are given in order that for any f  in L   1   the averages -1∑n - 1  i  n   i=0 T f  converge in the norm of L1   .

1. Introduction.

Let (Ω, A, μ)  be a σ  -finite measure space and let L1 = L1(Ω, A, μ)  denote the usual Banach space of all real-valued integrable functions on Ω  . A linear operator T : L1 →   L1   is called positive if f ≥  0  implies T f ≥ 0  , and a contraction if ∥T ∥1 ≤ 1  , with ∥T ∥1   denoting the operator norm of T  on L1   . We say that the pointwise ergodic theorem (resp. the L    1   -mean ergodic theorem) holds for T  if for any f  in L   1   the ergodic averages

           1 n∑-1  An(T )f =  --    Tif             n i=0

converge a.e. on Ω  (resp. in L1   -norm).

In 1964, Chacon [1] showed a class of positive contractions on L1   for which the pointwise ergodic theorem fails to hold and Ito [4] proved that if a positive contraction on L1   satisfies the L1   -mean ergodic theorem, then it satisfies the pointwise ergodic theorem. (cf. also Kim [5]).
More recently, Hasegawa and Sato [2] proved that if P1,...,Pd  are d  commuting positive contractions on L1   such that each Pi  , 1 ≤ i ≤ d  , satisfies the L1   -mean ergodic theorem, and if T1,...,Td  are (not necessarily commuting) contractions on L1   such that ∣Tif∣ ≤ Pi∣f∣ , for 1 ≤ i ≤ d  , then the averages

An(T1, ...,Td)f = An(T1) ...An(Td)f

converge a.e. for every f  in L   1   .

In view of these facts, it seems interesting to find out conditions on T  which guarantee the validity of the L1   -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 L1   and the space of all nonnegative functions in L∞ by L+1   and L+∞ , respectively. The adjoint operator of T  , which acts on L   (Ω,A, μ) = L    ∞              ∞ , is denoted by T * . Put

        ∞∑                           ∑∞  S ∞f =     T if, f ∈ L+1 ;    S*∞h  =     T *ih, h ∈ L+∞.          i=0                          i=0

By Hopf decomposition, Ω = C  ∪ D  , where C  and D  = Ω \ C  denote respectively the conservative and the dissipative part of Ω  respect to T  . We recall that C  and D  are determined uniquely mod μ  by:

  C1)   For all f ∈ L+1   , S ∞f  = ∞ a.e. on C ∩ {S∞f  >  0} , and
  D1)   For all f ∈ L+       1   , S ∞f  < ∞ a.e on D  .

For any A ∈ A  , χA  will be the characteristic function of the set A  . Let us write

             n-1       *     1-∑    *i  An(T  )h = n     T  h,     h ∈ L∞.               i=0

From the results of Helmberg [3] and Lin and Sine [7] about the relationship between the validity of the L1   -mean ergodic theorem for T  and the almost everywhere convergence of the averages      *  An(T  )h  , h ∈ L ∞ , we have:

Theorem 1.1. [Helmberg; Lin-Sine] Let T  be a positive contraction on L1   . Then the following are equivalent:
  1. The L1   -mean ergodic theorem holds for T  .
  2. For any h  in L    ∞ , the averages A  (T *)h    n  converge a.e.
  3.      *n  lim T   χD =  0  a.e. and there exists a nonnegative function f  in L1   satisfying T f = f  and {f  > 0}=  C  .

Recall that a subset K  of L1   is called weakly sequentially compact if every sequence {φn} of elements in K  contains a subsequence {φnk} which converges weakly to an element in L1   , that is, there exists φ  in L1   such that for any h  in L ∞

    ∫            ∫  lim    hφ   dμ =    hφ dμ.    k      nk

Kim [5] has proved:

Theorem 1.2. [Kim] Let T  be a positive contraction on L1   . Suppose that the sequence {An(T  )w}n   is weakly sequentially compact for some w >  0  in L1   . Then for each f  in L1   , lim An(T )f  exists in the L1   -norm and almost everywhere.

The purpose of this paper is to prove the following result:

Theorem A. Let T  be a positive contraction on L1   . Then the following assertions are equivalent:

  1. The L1   -mean ergodic theorem holds for T  .
  2. lim T *nχ  =  0          D  a.e. and there exists w  in L+    1   such that {w >  0}=  C  and the sequence {An(T  )w} is weakly sequentially compact.
  3. There exists w  in  +  L1   such that the averages An(T  )w  converge a.e. to a function w0   and lim T *nχ{w0=0} = 0  a.e.

2. The proofs.

We refer the reader to Krengel's book [6] for a proof of the following properties related with Hopf decomposition of Ω  :

  P1) For all h ∈ L+∞ , S*∞h  = ∞ a.e. on C ∩ {S *∞h >  0} .
  P2)  If      +  h ∈ L∞ and  *  T h ≤ h  , then  *  T h =  h  a.e. on C  .
  P3T *χD ≤  χD  a.e.

By P2   ) and P3   ) we see that T *χ  =  1      C  a.e. on C  .
We start with the following lemmas.

Lemma 2.1. Let f  be a function in  +  L1   such that the averages An(T )f  converge a.e. Let us denote the limit function by f0   . Then T f0 = f0   a.e.

Proof. Because of the identity

            n-+-1-            f-  T An(T )f =   n   An+1(T )f - n ,
we have lim T An(T )f = f0   a.e. Since f0 ∈ L1   , the sequence
An(T )f ∧ f0 :=  min{An(T  )f,f0}
converges to f0   in L1   -norm by Lebesgue's theorem. Then T (An(T )f ∧ f0)  converges to Tf0   in L1   -norm. It follows that T f0 ≤ f0   a.e. Then S∞(f0 -  Tf0) ≤ f0   a.e. and by statement C1   ) we see that f0 = T f0   a.e. on C  .

By D1   ) we have f  = 0   0  a.e. on D  and therefore Tf  = 0    0  on D  . The lemma is proved.

Lemma 2.2. Let f  in L+1   such that T f = f  and let C0 = {f >  0} , D0 =  {f = 0} . Then we have:
   i)     *  T  χD0 ≤  χD0   a.e.
  ii)   C0  ⊂ C  and T*χC0 =  χC0 + h  , with h ∈ L+∞ such that {h >  0}⊂  D  and S *∞h ≤  1  a.e.

Proof. Being T * a positive contraction on L    ∞ , i) follows from

     ∫            ∫                          *  0 =    χD0f  dμ =    fT  χD0 dμ.
Statement C1)  implies C0 ⊂  C  and from
∫         ∫            ∫                               *    f dμ =    f χC0 dμ =    fT  χC0 d μ,
we conclude that   *  T  χC0 = 1  a.e. on C0   . Then   *  T  χC0 = χC0 +  h  , with h  in   +  L ∞ and {h > 0} ⊂ D0   . From this, it is easy to see that for all n
                 n∑-1  T *nχ   =  χ  +     T *kh.       C0     C0                   k=0
Consequently,  *          *n  S∞h  = lim T  χC0 -  χC0 ≤ 1  a.e. and by property P1   ), {h >  0} ⊂ D  .

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)   lim T *nχD =  0  a.e. and there exists w  in L+1   such that {w  > 0}=  C  and the sequence {An(T )w} is weakly sequentially compact.
   ii) There exists w  in  +  L1   such that the averages An(T )w  converge a.e. to a function w0   and      *n  lim T   χ{w0=0} = 0  a.e.
  iii)  lim T *nχD =  0  a.e. and there exists w  in L+1   satisfying Tw =  w  and {w  > 0}=   C  .

The implications iii)⇒ ii) and iii)⇒ i) are immediate.

i)⇒ iii) By the mean ergodic theorem of Yosida and Kakutani [8], the sequence {An(T  )w} converges in L1   -norm to a function w0 ∈ L+1   such that T w0 = w0   . Put C0 =  {w0 > 0} and D0  = {w0 =  0} . By i) of lemma 2.2 there exists the a.e. lim T *nχ   = u          D0  and {u >  0} ⊂ D               0   . Since T *u = u  (see e.g. [3]) we have:

    ∫            ∫                        ∫                ∫  0 =    uw0 dμ =     ulim  An(T )w dμ = lim   uAn(T  )w dμ =    uw d μ

where the third equality follows from the fact that {An(T )w} being L1   -norm convergent is weakly convergent in L1   .
Then {u >  0} ⊂ D  . Moreover, from lim  T*nχD  = 0  a.e., we can see that u = 0  a.e.
On the other hand, from ii) of lemma 2.2 we obtain T*nχC  =  χC        0     0   on C  , for all n  . As T *nχ  =  1       C  on C  , we deduce that for all n  T *nχ     = χ       C\C0     C\C0   on C  . Therefore, for all n  ,            *n          *n  χC \C0 = T   χC\C0 ≤ T   χD0   a.e. on C  . Thus μ(C  \ C0) = 0  and iii) follows.

ii)⇒ iii) By lemma 2.1, T w  = w     0     0   . Put C  = {w   > 0}   0      0 and D  =  {w  = 0}    0     0 . By ii) of lemma 2.2, D ⊂  D0   . Then       *n  lim T   χD = 0  a.e. and      *n  lim T   χC\C0 = 0  a.e. Now, iii) follows as in i)⇒ iii). □

Remarks.
  1. In iii) of Theorem A, the condition lim T *nχ{w0=0} = 0  , can not be replaced by lim T *nχ  =  0          D  . To see this, take τ : Ω → Ω  an ergodic, conservative measure preserving transformation with respect to μ  , where μ  is σ  -finite and infinite. Then, the operator Tf =  f0τ  satisfies the pointwise ergodic theorem, but the L1   -mean ergodic theorem does not hold for T  .
  2. Suppose the L1   -mean ergodic theorem holds for T  . Then lim  T*nχD  = 0  a.e. For each g  in L    ∞ , we denote by g* the a.e. limit of A (T *)g   n  . Now, let h  in L∞ . Since      *                     *  ∣An(T  )(hχD) ∣ ≤ ∥h∥∞An(T   )χD  , we have
    h * = lim A  (T*)(hχ  ) = (hχ  )*.             n        C        C
    Put  *         *  hC = (hχC ) . Then, for all n
      *     *n *     *n     *     *n     *  h C = T   hC = T   (χC hC) + T  (χDh C ),
    and we conclude that  *        *n     *  h  = lim  T  (χC hC)  a.e.
In fact, we have:

Proposition 2.3. The following assertions are equivalent:
   i)  lim T *nχD =  0  a.e.
  ii)  Let u  in L    ∞ . Then T*u =  u  if and only if u = lim  T*n(χ  u)               C  and   *  T  (χC u) = χC u  on C  .

Sketch of proof.

i)⇒ ii) follows from T *χD ≤  χD  a.e. and the fact that u = lim T *n(χC u)  implies T *u = u  .

ii)⇒ i) let h =  lim T *nχ               D  . Then T *h = h  and {h > 0} ⊂  D  .

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

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