(1)THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM FOR BOUNDED BESICOVICH SEQUENCES K

THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM FOR BOUNDED BESICOVICH SEQUENCES

K. EL BERDAN

Abstract. In this paper we prove maximal ergodic theorem and a pointwise con- vergence theorem. Our result is to prove the convergence of

B_n(T, α, f) = 1 n

nX−1 j=0

α_jT^jf

for allf ∈ L¹(Ω, X) =L¹(X), wherentends to infinity, Ω is aσ-finite measure space,X is a reflexive Banach space,α_jis a bounded Besicovich sequence andT is a linear operator onL¹(X) which is contracting in bothL¹(X) andL^∞(X).

Our result has the additional advantage as it is sufficiently general in order to extend the Beck and Schwartz random theorem.

We can also generalize this result to a multidimensional case.

Notations and Definitions

Denote byX a Banach space, (Ω, β, µ) aσ-finite measure space,kxk_Xthe norm of a vectorxinX.

•L¹(Ω, X) = L¹(X) = n

f: Ω →X, measurable andR

Ωkf(ω)k_Xdµ(ω)<∞o the space of integrable functions in the sense of Bochner which take values inX, andL^∞(Ω, X) =L^∞(X) =n

f: Ω→X, measurable and bounded a.e.

(i.e supω∈Ωkf(ω)k_X<∞)o .

•L¹=L¹(Ω, R),L^∞=L^∞(Ω, R).

•For allf ∈L¹(X),kfk₁=R

Ωkf(ω)k_Xdµ(ω) andkfk_∞= sup_ω∈Ωkf(ω)k_X.

•For an operatorT ofL¹(X) into itself: T is contracting inL¹(X) iffkT fk₁≤ kfk₁ for all f ∈ L¹(X), similarly, T is contracting in L^∞(X) iff for all f ∈ L^∞(X),kT fk_∞≤ kfk_∞.

•For a > 0 and f ∈ L¹(X), f^a− = k^ffkmin{kfk, a}, f^a+ = f −f^a−, f^∗ = sup_nkBn(α, T, f)k, e^∗(a, α) = {f^∗ > αa}, e(a) = {kfk> a} and forA ⊂Ω we denoteϕA the indicator function ofA.

Received July 7, 1995; revised March 5, 1998.

1980Mathematics Subject Classification(1991Revision). Primary 27A35; Secondary 28A65.

We first define the term “Bounded Besicovich sequence”. Letαj be a sequence of complex numbers. We say thatαj is a bounded Besicovich sequence if

(i) there exists a positive realαsuch that|αj|< α for everyj∈N, (ii) forε >0 there exists a trigonometric polynomialϕεsuch that

limn

1 n+ 1

Xn j=0

|αj−ϕε(j)|< ε .

Introduction

In [5] J. Olsen proved an individual weighted ergodic theorem for bounded Besicovich sequences. He proved the a.e. convergence of

Bn(T, α, f) = 1 n+ 1

Xn j=0

αjT^jf

whereT is linear operator onL¹=L¹(Ω, R) which is contracting inL¹=L¹(Ω, R) and inL^∞=L^∞(Ω, R), andαj is a bounded Besicovich sequence.

In [2] R. V. Chacon proved a maximal ergodic lemma for operators which act in the space of functions taking their values in a Banach space, and he used this result to obtain a vector valued ergodic theorem as a generalization of the Dunford and Shwartz theorem.

In this paper we intend to generalize the individual ergodic theorem of Olsen to operators acting inL¹(X).

In his proof Olsen used the dominated operator of T-linear modulus (see [5, Lemma 2.1]) but in our case we have shown in [4] that there exists a vec- tor operator without linear modulus contracting inL¹.

We prove that the method of Chacon can be adapted to this situation, then our result is the convergenceµ-a.e. of Bn(T, α, f) whereT is a linear operator on L¹(X) which is contracting in bothL¹(X) andL^∞(X).

In the second part, we generalize these results to a multidimensional case: If αj is a bounded Besicovich sequence andλd=i1s1+· · ·+idsd wheresj ∈N for j= 1, . . . , dthen the limit of

Bn(T, d, α, f) = 1 (n+ 1)^d

Xn i1=0

· · · Xn

id

αλ_dT^λdf

exists a.e. for allf ∈L¹(X).

I. One Dimensional Case We now state Chacon’s result [2]:

Theorem 1.1 (Chacon). LetT be a linear operator on L¹(X) contracting in L¹(X)and in L^∞(X).

(i) Ifa >0,e^∗(a) = (

ω∈Ω; sup_n

1 n+ 1

Pn

j=0T^jf(ω) _X > a

)

Z

e^∗(a) a−f^a−(ω)_X

dµ(ω)≤Z

Ω

f^a+(ω)_Xdµ(ω).

(ii) Forf ∈L¹(X), the limit of An(T, f)(ω) = 1 n

nP−1

j=0T^jf(ω) exists strongly for everyω∈Ωas ntends to infinity.

(iii) If1< p <∞, then there exists a functionf^∗∗ ∈L^p(X)such that An(T)f_X ≤f^∗∗_X (a.e. n≥0) .

Main Result

We now state and prove our main result.

Theorem 1.2. LetX be a reflexive Banach space, T be a linear operator on L¹(X)contracting in L¹(X)and inL^∞(X),αj be a bounded Besicovich sequence then:

(i) For f ∈ L¹(X), the limit of Bn(T, α, f)(ω) = 1 n

nP−1

j=0αjT^jf(ω) exists strongly for everyω∈Ωasn tends to infinity.

(ii) If1< p <∞,f ∈L^p(X)the average Bn(T, α, f)converges a.e. and sup

n Bn(T, α, f)_{X p}≤

p p−1

1/p

kfk_p.

Before proving Theorem 1.2, let us remark that ifαj= 1 for everyj, we have e^∗(a,1) =e^∗(a) =

( supn

1 n+ 1

Xn k=0

T^kf > a

) .

We will have to prove that Chacon’s lemma is valid also for the averages Bn(T, α, f).

Lemma 1.3. If f ∈L¹(X)and a >0, we have then Z

e^∗(a,α) a−f^a−(ω)

X

dµ(ω)≤ Z

Ω

f^a+(ω)

Xdµ(ω).

Proof of Lemma 1.3. We can suppose thatαj >0 for allj inN. As in [2] we define:

f0=f^a+, fi+1=T fi− T fi

T f_iXmin

(T f_iX, a−f^a−_X− Xi k=0

T f_k−fk+1_X) and

d0= 0, di+1 = T fi

T fi

X

min

(T fi_X, a−f^a−_X− Xi k=0

T fk−fk+1_X) .

By [2] these sequences satisfy the following relations:

f^a−(ω)_X+ Xi k=0

dk(ω)_X≤afor everyi inN and for everyω in Ω, (1)

T fi(ω)_X=fi+1(ω)_X+di+1(ω)_X for allω in Ω, (2)

Tⁱf =Tⁱf^a−+fi+ Xi k=0

T^i−kdk for everyiinN, (3)

Xn i=0

hTⁱf^a−+ Xi k=0

T^i−kdk

i= Xn

i=0

Tⁱ f^a−+

n−i

X

k=0

dk

for everyninN, (4)

if forω∈Ωfi+1(ω)6= 0 thena=f^a−(ω)_X+ Xi+1 k=0

dk(ω)_X. (5)

Multiply equality (3) byαj to obtain αjTⁱf =αiTⁱf^a−+αifi+

Xi k=0

αiT^i−kdk.

To prove the following equality (∗)

Xn i=0

hαiTⁱf^a−+ Xi k=0

αiT^i−kdk

i= Xn i=0

Tⁱ

αif^a−+

n−i

X

k=0

αi+kdk

we shall argue by induction onn. It is true forn= 0. Suppose now that it is valid up tonand let us prove it forn+ 1:

n+1X

i=0

hαiTⁱf^a−+ Xi k=0

αiT^i−kdk

i

= Xn i=0

αi

Tⁱf^a−+ Xi k=0

T^i−kdk

+αn+1

Tⁿ⁺¹f^a−+

n+1X

k=0

T^n+1−kdk

= Xn i=0

Tⁱ

αif^a−+

n−i

X

k=0

αi+kdk

| {z }

µn

+αn+1

Tⁿ⁺¹f^a−+

n+1X

k=0

T^n+1−kdk

| {z }

λn

.

On the other hand we have

n+1X

i=0

Tⁱ

αif^a−+

n+1X−i k=0

αidk

= Xn i=0

Tⁱ αif+

n−i

X

k=0

αi+kdk+αn+1dn+1−i

= Xn i=0

Tⁱ αif+

n−i

X

k=0

αi+kdk

+Tⁿ⁺¹f^a−+αn+1 n+1X

i=0

Tⁱdn+1−i

= Xn i=0

Tⁱ αif^a−+

n−i

X

k=0

αi+kdk

!

| {z }

µn

+αn+1 Tⁿ⁺¹f^a−+

n+1X

k=0

T^n+1−kdk

!

| {z }

λ_n

.

It follows that (∗) holds for everyninN. Thus

n+1X

i=0

αiTⁱf = Xn i=0

Tⁱ

αif^a−+

n−i

X

k=0

αi+kdk

+

n+1X

i=0

αifi

=

α0f^a−+ Xn k=0

αkdk

+ Xn i=1

Tⁱ

αif^a−+

n−i

X

k=0

αi+kdk

+

n+1X

i=0

αifi

Now, we shall show that for allω∈e^∗(α, a)=n

ω∈Ω; sup

n

1 n

Pn

j=0αjT^jf(ω)_X> ao we have

a=f^a−(ω)_X+ X∞ k=0

dk(ω)_X.

Letω∈e^∗(α, a) then there existsn=n(ω) such that:

αan+αa≤

nX−1 i=1

αiTⁱf(ω)≤

α0f^a−(ω)_X+ Xn k=0

αkd_k(ω)_X +

Xn i=1

Tⁱ

αif^a−+

n−i

X

k=0

αi+kdk

(ω)_X+ Xn i=0

αifi(ω)_X.

By (1) we have αif^a−(ω)

X+ Xn k=0

αi+kd_k(ω)

X ≤αif^a−(ω)

X+ Xn k=0

αid_k(ω)

X

=αihf^a−(ω)_X+ Xn k=0

dk(ω)_Xi and, as the operatorT is contracting inL^∞(X), then

Xn i=1

Tⁱ(αif^a−+

n−i

X

k=0

αi+kdk(ω)_X ≤ Xn i=1

Tⁱ(αif^a−+

n−i

X

k=0

αi+kdk)_∞

≤ Xn i=1

αif^a−+

n−i

X

k=0

αi+kdk_∞= Xn i=1

ωsup∈Ω

αif^a−+

n−i

X

k=0

αi+kdk

(ω)_X

≤ Xn i=1

ωsup∈Ω

hαif^a−(ω)_X+

n−i

X

k=0

αi+kd_k(ω)_X

≤αai whence

(n+a)αa≤αhf^a−(ω)_X+ Xn k=0

dk(ω)_Xi

+naα+

nX−1 i=1

αifi(ω)_X

≤αnhf^a−(ω)_X+ Xn k=0

dk(ω)_Xi

+na+

n−1

X

i=1

αifi(ω)_Xo and so

a≤hf^a−(ω)_X+ Xn k=0

d_k(ω)_Xi +

nX−1 i=1

f_i(ω)_X

by relation (5) we have

a=f^a−(ω)_X+ X∞ k=0

dk(ω)_X for allω∈e^∗(α, a).

The rest of the proof can be obtained in the same way as in Chacon’s method, in fact, using (6) and knowing thatkTk₁≤1 we deduce

Z

e^∗(a,α) a−f^a−(ω)_X dµ≤

X∞ k=0

T fk₁−fk+1₁≤ X∞ k=0

fk₁−fk+1₁

≤f₀₁=Z

Ω

f₀(ω)dµ(ω) =Z

Ω

f^a+(ω)dµ(ω).

Proof of Theorem1.2. Since the Lemma 1.3 gives us maximal weak inequality for averages Bn(T, α, f) it suffices to prove the convergence forf belonging to a set which is dense every where inL¹(X). We know thatL^∞(X) is such a set, so forf ∈L^∞(X) we have:

1 n+ 1

Xn i=0

αiTⁱf = 1 n+ 1

Xn i=0

ϕε(i)Tⁱf+ 1 n+ 1

Xn i=0

αi−ϕε Tⁱf.

Letθ be a complex number. The operatorUf =e^iθT f is contracting in both L¹(X) and L^∞(X) and the theorem follows in the caseαn=e^inθ from Chacon’s theorem. The linearity of convergence gives that

limn

1 n+ 1

Xn i=0

ϕε(i)Tⁱf

exists and is finite a.e. for any trigonometric polynomialϕε, andf ∈L^∞(X).

In fact we have for this operators a strong inequality inL^∞(X):

sup

n

1 n+ 1

Xn i=0

ϕε(i)Tⁱf_X

_∞≤kεkfk_∞ and by the definition b) we also have

lim sup

n

1 n+ 1

Xn i=0

|αi−ϕε(i)|Tⁱf_X≤εkfk_∞. By Lemma 1.3 we have

aµ(e^∗(a, α))≤ Z

e(a)

kf(ω)k_Xdµ and so, using the rearrangement formula we get:

f^∗P_p =Z

Ω[f^∗]^pdµ=pα^pZ

Ω

Z f^∗/α

0 λ1e^∗(λ,α)dλ dµ(ω)

=pα^p Z

Ω

Z ^∞

0 λ^p−1µ

e^∗(λ, α) dλ

≤pα^pZ

Ω

Z ^∞

0 λ^p−2f(ω)_Xdλ dµ(ω)

=pα^pZ

Ω

Z ^∞

0 λ^P−21e(λ)dλ dµ(ω)

=pα^pZ

Ω

f(ω)_XhZ ^kf(ω)k

0 λ^p−2dλi dµ(ω)

=α^p p p−1

Z

Ω

f(ω)^p_Xdµ(ω).

Remark 1.4. We notice that Lemma 1.3 remains true for any bounded se- quence even if it is not a Besicovich one.

Let us consider some examples to which Theorem 1.2 is applied:

Examples 1.5.

1. Let X = R×R (reflexive Banach space) with normk(x, y)k= kx|+|y| . Ω ={1,2}a probability space,µ(1) =µ(2) = ¹₂;L¹({1,2},R×R) being a Banach space of dimension 4. Notice that for

T =







a b a⁰ b⁰ c d c⁰ d⁰ e f e⁰ f⁰ g h g⁰ h⁰







T will be contracting onL¹({1,2},R×R) if the sum of the absolute values of the terms in each column is less than 1.

It is easy to show that the operatorT is contracting onL^∞({1,2},R×R) if the terms of the matrixT satisfy













|a|+|c|+|a⁰|+|c⁰| ≤1

|a|+|c|+|b⁰|+|d⁰| ≤1

|b|+|d|+|a⁰|+|c⁰| ≤1

|b|+|d|+|b⁰|+|d⁰| ≤1

and













|e|+|g|+|e⁰|+|g⁰| ≤1

|e|+|g|+|f⁰|+|h⁰| ≤1

|f|+|h|+|e⁰|+|g⁰| ≤1

|f|+|h|+|f⁰|+|h⁰| ≤1 LetT be the linear operator onL¹({1,2},R×R) represented by a square matrix of order 4 defined by

T =







2/9 0 3/8 3/7 1/4 1/9 2/9 2/9 0 1/7 2/7 1/4 2/5 1/9 2/9 3/10







T is contracting on bothL¹({1,2},R×R) andL^∞({1,2},R×R).

Consider also the sequenceαn =e^iθnwhere θ a complex number, we have by Theorem 1.2 the convergence of the sequence

1 n+ 1

Xn k=0

αkT^kf = 1 n+ 1

Xn k=0

e^iθk







2/9 0 3/8 3/7 1/4 1/9 2/9 2/9 0 1/7 2/7 1/4 2/5 1/9 2/9 3/10







k



 x1

y1

x2

y2







for allf = x1

y1

ϕ{1}+

x2

y2

ϕ{2}∈L¹({1,2},R×R).

2. LetX be a reflexive Banach space, Ω a probability space,ϕis a transforma- tion from Ω to Ω such that for allf ∈L¹(X),

Z

Ω

f ◦ϕ(ω)dµ(ω)≤ Z

Ω

f(ω)dµ(ω)

(or ϕis a measure preserving transformation) and let (αi)i∈N be any Besicovich bounded sequence then

limn

1 n+ 1

Xn i=0

αi(f◦ϕⁱ) exists a.e. for everyf ∈L¹(X).

Applications. The general result of Theorem 1.2 can be applied to give a gen- eralization of the vector valued random ergodic theorem of Beck and Schwartz [1].

Theorem 1.6. Let be defined onΩa strongly measurable functionUωwith val- ues in the Banach spaceB(X)of bounded linear operators on a spaceX. Suppose that kUωk ≤ 1 for all ω ∈ Ω. Let ϕ be a measure preserving transformation in (Ω, β, µ) and(αi)i∈N be a Besicovich bounded sequence, then forf ∈L¹(X), the limit

limn

1 n+ 1

Xn k=0

αkUωUφ(ω). . . Uφ^k−1(ω)f(ϕ^k(ω)) exists for almost allω∈Ω.

Proof. Forf ∈L¹(X) we define

Uf(ω) =Uω(f(ϕ(ω))).

Then it can be easily seen that it satisfies the conditions of Theorem 1.2 and hence

the condition follows at once from Theorem 1.2.

II. A Multidimensional Case

Obtaining an extension of Theorem 1.2 to distinct several operatorsT1, . . . , Td

which more general means difficult. But ifT1=T^s1,. . . . Td=T^sd whereT is an linear operator onL¹(X) and sk ∈N for k= 1, . . . , d, then Theorem 1.2 can be extended to this case. Let

Bn(T, d, α, f) = 1 (n+ 1)^d

Xn i1=0

· · · Xn id=0

αλ_dT₁ⁱ¹. . . T_d^idf

where λd = i1s1+. . . idsd and αj be a bounded Besicovich sequence. Let f_d^∗ = sup_nBn(T, d, α, f)_X ande^∗_d(a, α) =

f_d^∗> αa .

Theorem 2.1. Let X be a reflexive Banach space, T be a linear operator on L¹(X)contracting inL¹(X)and inL^∞(X),αj be a bounded Besicovich sequence.

Then forλd =i1s1+· · ·+idsd,s=s1+. . . , sd: (i) For f ∈L¹(X), eta >0we have

Z

e^∗_d(a,α) a−f^a−(ω)_X

dµ(ω)≤sZ

Ω

f^a+(ω)_Xdµ(ω).

(ii) For f ∈ L¹(X), the limit of Bn(T, d, α, f)(ω) exists strongly for every ω∈Ωasn tends to infinity.

(iii) If 1< p < ∞,f ∈L^p(X) andα = sup_kαk, the average Bn(T, d, α, f) converges a.e. and

sup

n B_n(T, d, α, f)_Xp≤α p

p−1 1/p

kfk_p.

Proof. We will prove the theorem in the case where d= 2, and s1 = s2 = 1 only, for the sake of simplicity. A similar proof to that used in this case gives the general result in thed-dimensional case (d >2).

We now study the following averages:

Bn(T,2, α, f)(ω) = 1 (n+ 1)²

Xn i=0

Xn j=0

αi+jT^i+jf.

By the relation (3) in the proof of the Theorem 1.2 we can write Xn

i=0

Xn j=0

αi+jT^i+jf = Xn j=0

Xn k=0

αj+k

hT^j+kf^a−+

j+kX

m=0

T^j+k−mdm

i+ Xn j=0

Xn k=0

αj+kf^j+k.

We prove an analogous equality to (∗) as in the proof of Theorem 1.2

Xn j=0

Xn k=0

αj+k

hT^j+kf^a−+

j+kX

m=0

T^j+k−mdm

i= Xn j=0

n+jX

t=j

αt

T^tf^a−+ Xt m=0

T^t−mdm

| {z }

1

(7)

= Xn j=0

n^n+jX

t=0

αt T^tf^a−+ Xt m=0

T^t−mdm

−

j−1

X

t=0

αt

T^tf^a−+ Xt m=0

T^t−mdm

o

| {z }

1

= Xn j=0

n^n+jX

t=0

T^t

αtf^a−+

n+jX−t m=0

αt+mdm

| {z }

2

−

j−1

X

t=0

T^t

αtf^a−+

j−X1−t m=0

αt+mdm

o by (∗)

= Xn j=0

n^n+jX

t=0

T^th

αtf^a−+

j−Xt−1 m=0

αt+mdm+

n+jX−t m=j−t

αt+mdm

| {z }

2

i

−

j−1

X

t=0

T^t

αtf^a−+

j−X1−t m=0

αt+mdm

o

= Xn j=0

n^n+jX

t=0

T^t

αtf^a−+

j−Xt−1 m=0

αt+mdm

+

n+jX

t=0

h^n+jX^−t

m=j−t

αt+mdm

| {z }

3

i

−

j−1

X

t=0

T^t

αtf^a−+

j−X1−t m=0

αt+mdm

o

= Xn j=0

n+jX

t=j

T^t

αtf^a−+

j−Xt−1 m=0

αt+mdm

+

n+jX

t=0

T^tn+jX^−t

m=j−t

αt+mdm

| {z }

3

o.

Let

χn(ω) = Xn j=0

Xn k=0

αj+k

hT^j+kf^a−+

j+kX

m=0

T^j+k−mdm

i(ω)

and

f_C^∗ = sup

n Cn(T, α, f)_X where

Cn(T, α, f) = 1 (2n+ 1)²

Xn j=0

Xn k=0

αj+kT^j+kf

ande^∗_C(a, α) ={f_C^∗ > aα}. Fixω∈e^∗_C(a, α) there existsn=n(ω) such that

(2n+ 1)²αa≤ Xn j=0

Xn k=0

αj+kT^j+kf(ω)_X≤χn(ω)_X+ Xn j=0

Xn k=0

αj+kfj+k_X.

But

χn(ω)_X ≤h

α0f^a−(ω)_X+ Xn k=0

αkdk(ω)_Xi +

Xn j=0

n+jX

t=j

T^t

αtf^a−+

j−Xt−1 m=0

αt+mdm

(ω)_X+ Xn j=0

n+jX

t=0

T^tn+jX^−t

m=j−t

αt+mdm

(ω)_X.

We know thatT is contracting inL^∞(X), using (1) we obtain:

χn(ω)_X ≤h

α0f^a−(ω)_X+ Xn k=0

αkdk(ω)_Xi +

Xn j=0

n+jX

t=j

aα+ Xn j=0

n+jX

t=0

aα

≤h

α0f^a−(ω)_X+ Xn k=0

αkdk(ω)_Xi +

Xn j=0

(n−j−j)αa+ Xn j=0

(n+j)αa . By (3) we can write

(2n+ 1)²αa≤h

α0f^a−(ω)_X+ Xn k=0

αkdk(ω)_Xi +n(n+ 1)αa+ 2n²αa+

Xn j=0

Xn k=0

αj+kfj+k_X

≤αhf^a−(ω)_X+ Xn k=0

dk(ω)_Xi

+ (3n²+n)αa+α Xn j=0

Xn k=0

fj+k_X.

As in the cased= 1 we have by the relations (3), (4) and (5) a≤hf^a−(ω)

X+ Xn k=0

dK(ω)

X

i+ X∞ j=0

X∞ k=0

fj+k(ω)

X. By the (5) we see for allω∈e^∗_C(a, α)

a=f^a−(ω)_X+ X∞ k=0

dk(ω)_X. This implies

Z

e^∗_C(a,α) a−f^a−(ω)_X

dµ(ω)≤ X∞ k=0

dk₁≤f0₁=Z

Ω

f0(ω)dµ(ω).

On the other hand we can write

Bn(T, α, f) = (2n+ 1)²

(n+ 1)² Cn(T, α, f)

which givesf_B^∗ ≤2f_C^∗ hencee^∗_B(a, α)⊆e^∗_C(a/2, α). For b=a/2 we have Z

e^∗_B(a,α) a−f^a−(ω)

X

dµ(ω)≤2Z

e^∗_C(a,α) b−f^b−(ω)

X

dµ(ω)

≤2Z

Ω

f^b+(ω)

Xdµ(ω)≤2Z

Ω

f^a+(ω)

Xdµ(ω) so

(8) aµ

e^∗_B(a, α)≤aµ

e^∗_C(a/2, α)≤2Z

Ω

f(ω)dµ(ω).

Now, we shall prove that averages B(n1, n2, T, α, f) = 1

n1n2 n1

X

i=0 n2

X

j=0

ϕ¹(i)ϕ²(j)T^i+jf converge on a dense set inL¹(X). We will need the following lemma:

Lemma 2.2. LetT be a linear operator onL¹(X)which is contracting in both L¹(X)andL^∞(X), then forf ∈Inv (T) + Im (I−T)∩L^∞(X)the limit

lim 1 n1n2

n1

X

i=0 n2

X

j=0

T^i+jf

exists asn1 andn2 tend to infinity.

Proof. Letf =g+ (h−T h) withT g=g,g∈L¹(X) andh∈L^∞(X). Then 1

n1n2 n1

X

i=0 n2

X

j=0

T^i+jf =g+ 1 n1n2

n1

X

i=0

TⁱhXⁿ²

j=0

(T^jh−T^j+1f)i

=g+ 1 n1n2

n1

X

i=0

Tⁱh−Tⁿ²⁻¹fi

=g+ 1 n1

h 1 n2

n₁

X

i=0

Tⁱhi

− 1 n1

h 1 n2

n₁

X

i=0

Tⁿ²⁻¹⁺ⁱhi .

ButT_∞≤1 hence 1

n2

h 1 n1

n₁

X

i=0

Tⁱhi_∞≤ 1

n2h_∞−−−−→^n2→∞ 0

and h 1

n2

h 1 n1

n₁

X

i=0

Tⁿ²⁻¹⁺ⁱhi_∞≤ 1 n2

h_∞−−−−→^n2→∞ 0.

LetUkf =e^iθkTkf, k= 1,2. Uk is a linear operator satisfying the conditions of Lemma 2.2 and so the theorem holds in the caseφ^k(n) =e^θkn.

This implies that _n1¹_n2Pn1

i=0Pn2

j=0ϕ¹(i)ϕ²(j)T^i+jf converges a.e. on Inv (T) + Im (I−T)∩L¹(X) which is dense, by Kakutani-Yoshida theorem inL¹(X) (X a reflexive Banach space). From Theorem 2.1(i) and the linearity of convergence of sequences we obtain that

n1,nlim2→∞

1 n1n2

n₁

X

i=0 n₂

X

j=0

ϕ¹(i)ϕ²(j)T^i+jf

exists and is finite a.e. for any trigonometric polynomial φ^k, k = 1,2, and f ∈ L¹(X).

The general case (d >2) is similar to the real case studied by Olsen [5].

The second assertion (ii) follows from the maximal equality (3) and the re- arrangement formula used in part I.

Letαj = 1, andsk= 1 for alljinN andk= 1, . . . , dthe averageBn(T, d, α, f) becomes

Bn(T, d, α, f) = 1 (n+ 1)^d

Xn i₁=0

· · · Xn i_d=0

T^i1+···+idf = 1 n+ 1

Xn j=0

T^jd

f.

Using Theorem 2.1, we deduce the following:

Corollary 2.3. LetX be a reflexive Banach space,T be a linear operator on L¹(X)contracting in L¹(X) and inL^∞(X), then ford∈N andf ∈L¹(X)

limn

1 n+ 1

Xn j=0

T^jd

f

exists a.e.

Acknowledgement. The author wishes to express his thank to Professors Sylvie Delabriere, Louis Sucheston, Jean Paul Thouvenot and the referee for kind advice.

References

1.Beck A. and Shwartz J. T.,A vector valued random theorem, Proc. Amer. Math. Soc.8 (1957), 1049–1059.

2.Chacon R. V.,A vector ergodic theorem for operators satisfying norm conditions, J. Math.

11(1962), 165–172.

3.Berdan K. El,Theoreme ergodique vectoriel pour les transformations ponctuelles preservant la mesure sur, C. R. A. S. Paris t. 317, serie I.

4. ,These d’Universite Paris 6. Theoremes ergodiques a plusieurs parametres dans les espaces de Banach, 1995.

5.Olsen J.,The individual weighted ergodic theore for bounded Besicovich sequences, Canad.

Math. Bull.25, 40(1982).

K. El Berdan, Lebanese University, Faculty of Sciences I, Departement of Mathematics, Hadeth- Beirut, (Mazraa P. O. Box: 14 - 6 573), Lebanon;e-mail: [email protected]