1
Acta Math. Univ. Comenianae
Vol. LXVIII, 1(1999), pp. 1–15
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 convergence theorem. Our result is to prove the convergence of
Bn (T, α, f ) =
n−1
1 X
αj T j f
n j=0
for all f ∈ L1 (Ω, X) = L1 (X), where n tends to infinity, Ω is a σ-finite measure
space, X is a reflexive Banach space, αj is a bounded Besicovich sequence and T is
a linear operator on L1 (X) which is contracting in both L1 (X) and L∞ (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 by X a Banach space, (Ω, β, µ) a σ-finite measure space, kxkX the norm
of a vector x in X.
n
o
R
• L1 (Ω, X) = L1 (X) = f : Ω → X, measurable and Ω kf (ω)kX dµ(ω) < ∞
the space of integrable functions in
n the sense of Bochner which take values
∞
∞
in X, and L (Ω, X) = L (X) = f : Ω → X, measurable and bounded a.e.
o
(i.e supω∈Ω kf (ω)kX < ∞) .
• L1 = L1 (Ω, R), L∞ = L∞ (Ω,
R R).
1
• For all f ∈ L (X), kf k1 = Ω kf (ω)kX dµ(ω) and kf k∞ = supω∈Ω kf (ω)kX .
• For an operator T of L1 (X) into itself: T is contracting in L1 (X) iff kT f k1 ≤
kf k1 for all f ∈ L1 (X), similarly, T is contracting in L∞ (X) iff for all f ∈
L∞ (X), kT f k∞ ≤ kf k∞ .
• For a > 0 and f ∈ L1 (X), f a− = kff k min{kf k, a}, f a+ = f − f a− , f ∗ =
supn kBn (α, T, f )k, e∗ (a, α) = {f ∗ > αa}, e(a) = {kf k > a} and for A ⊂ Ω
we denote ϕA the indicator function of A.
Received July 7, 1995; revised March 5, 1998.
1980 Mathematics Subject Classification (1991 Revision). Primary 27A35; Secondary 28A65.
2
K. EL BERDAN
We first define the term “Bounded Besicovich sequence”. Let αj be a sequence
of complex numbers. We say that αj is a bounded Besicovich sequence if
(i) there exists a positive real α such that |αj | < α for every j ∈ N ,
(ii) for ε > 0 there exists a trigonometric polynomial ϕε such that
n
1 X
lim
|αj − ϕε (j)| < ε .
n n+1
j=0
Introduction
In [5] J. Olsen proved an individual weighted ergodic theorem for bounded
Besicovich sequences. He proved the a.e. convergence of
n
Bn (T, α, f ) =
1 X
αj T j f
n + 1 j=0
where T is linear operator on L1 = L1 (Ω, R) which is contracting in L1 = L1 (Ω, R)
and in L∞ = L∞ (Ω, R), and αj 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 in L1 (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 vector operator without linear modulus contracting in L1 .
We prove that the method of Chacon can be adapted to this situation, then
our result is the convergence µ-a.e. of Bn (T, α, f ) where T is a linear operator on
L1 (X) which is contracting in both L1 (X) and L∞ (X).
In the second part, we generalize these results to a multidimensional case: If
αj is a bounded Besicovich sequence and λd = i1 s1 + · · · + id sd where sj ∈ N for
j = 1, . . . , d then the limit of
Bn (T, d, α, f ) =
n
n
X
X
1
·
·
·
αλd T λd f
(n + 1)d i =0
i
1
exists a.e. for all f ∈ L1 (X).
d
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
3
I. One Dimensional Case
We now state Chacon’s result [2]:
Theorem 1.1 (Chacon). Let T be a linear operator on L1 (X) contracting in
L1 (X) and in L∞ (X).
(
)
1 P
n
T j f (ω) > a
(i) If a > 0, e∗ (a) = ω ∈ Ω; supn n + 1 j=0
X
Z
e∗ (a)
a − f a− (ω)X dµ(ω) ≤
Z
Ω
a+ f (ω) dµ(ω).
X
(ii) For f ∈ L1 (X), the limit of An (T, f )(ω) =
P j
1 n−1
T f (ω) exists strongly
n j=0
for every ω ∈ Ω as n tends to infinity.
(iii) If 1 < p < ∞, then there exists a function f ∗∗ ∈ Lp (X) such that
An (T )f ≤ f ∗∗ X
X
(a.e. n ≥ 0) .
Main Result
We now state and prove our main result.
Theorem 1.2. Let X be a reflexive Banach space, T be a linear operator on
L1 (X) contracting in L1 (X) and in L∞ (X), αj be a bounded Besicovich sequence
then:
P
1 n−1
αj T j f (ω) exists
(i) For f ∈ L1 (X), the limit of Bn (T, α, f )(ω) =
n j=0
strongly for every ω ∈ Ω as n tends to infinity.
(ii) If 1 < p < ∞, f ∈ Lp (X) the average Bn (T, α, f ) converges a.e. and
1/p
p
supBn (T, α, f ) ≤
kf kp .
n
X
p−1
p
Before proving Theorem 1.2, let us remark that if αj = 1 for every j, we have
)
n
1 X
sup T kf > a .
n n + 1
(
e∗ (a, 1) = e∗ (a) =
k=0
We will have to prove that Chacon’s lemma is valid also for the averages
Bn (T, α, f ).
4
K. EL BERDAN
Lemma 1.3. If f ∈ L1 (X) and a > 0, we have then
Z
Z
a − f a− (ω)X dµ(ω) ≤ f a+ (ω)X dµ(ω) .
e∗ (a,α)
Ω
Proof of Lemma 1.3. We can suppose that αj > 0 for all j in N . As in [2] we
define:
f0 = f a+ ,
fi+1
(
)
i
X
a− T fi
T fi , a − f −
T fk − fk+1 = T fi − X
X
X
T fi min
X
k=0
and
d0 = 0 ,
di+1
(
)
i
X
a− T fi
T fi , a − f −
T fk − fk+1 =
.
X
X
X
T fi min
X
k=0
By [2] these sequences satisfy the following relations:
(1)
i
X
a− f (ω) +
dk (ω) ≤ a for every i in N and for every ω in Ω,
X
X
(2)
T fi(ω) = fi+1 (ω) + di+1 (ω) for all ω in Ω,
X
X
X
(3)
T if = T if a− + fi +
k=0
i
X
T i−k dk for every i in N ,
k=0
(4)
n h
X
T if a− +
i=0
(5)
i
X
n
n−i
i X
X
T i−k dk =
T i f a− +
dk for every n in N ,
i=0
k=0
k=0
i+1
X
dk (ω) .
if for ω ∈ Ω fi+1 (ω) 6= 0 then a = f a− (ω)X +
X
k=0
Multiply equality (3) by αj to obtain
αj T if = αi T i f a− + αi fi +
i
X
αi T i−k dk .
k=0
To prove the following equality
(∗)
n h
i
n
n−i
i X
X
X
X
αi T if a− +
αi T i−k dk =
T i αi f a− +
αi+k dk
i=0
k=0
i=0
k=0
5
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
we shall argue by induction on n. It is true for n = 0. Suppose now that it is valid
up to n and let us prove it for n + 1:
n+1
Xh
αi T i f a− +
i=0
i
X
αi T i−k dk
i
k=0
=
n
X
αi T i f a− +
i=0
=
n
X
i
X
n+1
X
T i−k dk + αn+1 T n+1 f a− +
T n+1−k dk
k=0
T i αi f a− +
i=0
|
{z
n−i
X
k=0
αi+k dk + αn+1 T n+1 f a− +
k=0
}
µn
|
{z
n+1
X
T n+1−k dk .
k=0
}
λn
On the other hand we have
n+1
n+1−i
n
n−i
X
X X
X
T i αi f a− +
αi dk =
T i αi f +
αi+k dk + αn+1 dn+1−i
i=0
i=0
k=0
=
n
X
T i αi f +
i=0
=
n
X
n−i
X
αi+k dk + T n+1 f a− + αn+1
k=0
T
i
αi f
a−
i=0
+
n−i
X
n+1
X
αi+k dk
+ αn+1
}
µn
T i dn+1−i
i=0
!
k=0
{z
|
k=0
T
n+1 a−
f
+
{z
|
n+1
X
!
T
n+1−k
dk
k=0
.
}
λn
It follows that (∗) holds for every n in N . Thus
n+1
X
i=0
αi T i f =
n
X
n−i
n+1
X
X
T i αi f a− +
αi+k dk +
αi fi
i=0
i=0
k=0
n
n
n−i
X
n+1
X
X
X
= α0 f a− +
αk dk +
T i αi f a− +
αi+k dk +
αi fi
i=1
k=0
i=0
k=0
1 P
o
n
Now, we shall show that for all ω ∈ e∗ (α, a) = ω ∈ Ω; sup
αj T j f (ω) > a
n j=0
X
n
we have
∞ X
a− a = f (ω) +
dk (ω) .
n
X
k=0
X
Let ω ∈ e∗ (α, a) then there exists n = n(ω) such that:
n
n−1
X
X
αan + αa ≤ αi T i f (ω) ≤ α0 f a− (ω)X +
αk dk (ω)X
i=1
+
k=0
n X
i
T αi f a− +
i=1
n−i
X
k=0
n
X
αi+k dk (ω) +
αi fi (ω)X .
X
i=0
6
K. EL BERDAN
By (1) we have
n
n
X
X
αi f a− (ω)X +
αi+k dk (ω)X ≤ αi f a− (ω)X +
αi dk (ω)X
k=0
k=0
n
h
X
i
a−
dk (ω)
= αi f (ω) X +
X
k=0
∞
and, as the operator T is contracting in L (X), then
n n−i
X
X
i
αi+k dk (ω)
T (αi f a− +
i=1
X
k=0
≤
n
X
∞
k=0
n n−i
X
X
i
αi+k dk )
T (αi f a− +
i=1
n n−i
X
X
αi+k dk αi f a− +
i=1
≤
≤
n
X
=
∞
k=0
n−i
X
sup αi f a− +
αi+k dk (ω)
i=1 ω∈Ω
k=0
X
n−i
h X
sup αi f a− (ω)X +
αi+k dk (ω)X
i=1 ω∈Ω
i
≤ αa
k=0
whence
n
n−1
h
X
X i
dk (ω) + naα +
αi fi (ω)X
(n + a)αa ≤ α f a− (ω)X +
X
i=1
k=0
n
n−1
nh
X
X i
o
a−
≤α
αi fi (ω)X
f (ω) X +
dk (ω) X + na +
i=1
k=0
and so
n
h
X
X
i n−1
dk (ω) +
fi (ω)
a ≤ f a− (ω)X +
X
X
i=1
k=0
by relation (5) we have
∞
X
dk (ω)
a = f a− (ω)X +
X
k=0
for all ω ∈ 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 that kT k1 ≤ 1 we deduce
Z
∞
∞
X
X
T fk − fk+1 ≤
fk − fk+1 a − f a− (ω)X dµ ≤
1
1
1
1
e∗ (a,α)
k=0
≤ f0 1 =
Z
Ω
k=0
f0 (ω) dµ(ω) =
Z
Ω
a+ f (ω) dµ(ω).
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
7
Proof of Theorem 1.2. Since the Lemma 1.3 gives us maximal weak inequality
for averages Bn (T, α, f ) it suffices to prove the convergence for f belonging to a
set which is dense every where in L1 (X). We know that L∞ (X) is such a set, so
for f ∈ L∞ (X) we have:
n
n
n
1 X
1 X
1 X
αi T if =
ϕε (i)T i f +
αi − ϕε T i f.
n + 1 i=0
n + 1 i=0
n + 1 i=0
Let θ be a complex number. The operator U f = eiθ T f is contracting in both
L1 (X) and L∞ (X) and the theorem follows in the case αn = einθ from Chacon’s
theorem. The linearity of convergence gives that
n
1 X
lim
ϕε (i)T i f
n n+1
i=0
exists and is finite a.e. for any trigonometric polynomial ϕε , and f ∈ L∞ (X).
In fact we have for this operators a strong inequality in L∞ (X):
n
1 X
i ϕε (i)T f ≤ kε kf k∞
sup
n n+1
X
i=0
∞
and by the definition b) we also have
lim sup
n
n
1 X
|αi − ϕε (i)|T if X ≤ εkf k∞ .
n + 1 i=0
By Lemma 1.3 we have
Z
∗
aµ(e (a, α)) ≤
e(a)
kf (ω)kX dµ
and so, using the rearrangement formula we get:
Z Z f ∗ /α
Z
∗ P
f = [f ∗ ]p dµ = pαp
λ1e∗ (λ,α) dλ dµ(ω)
p
Ω
Ω 0
Z Z ∞
λp−1 µ e∗ (λ, α) dλ
= pαp
Ω 0
Z Z ∞
≤ pαp
λp−2 f (ω)X dλ dµ(ω)
Ω 0
Z Z ∞
= pαp
λP −2 1e(λ) dλ dµ(ω)
Ω
Z
0
Z
h kf (ω)k p−2 i
= pαp f (ω)X
λ
dλ dµ(ω)
0
Ω
Z
p
f (ω)p dµ(ω).
= αp
X
p−1 Ω
8
K. EL BERDAN
Remark 1.4. We notice that Lemma 1.3 remains true for any bounded sequence 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 norm k(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

a b a0
 c d c0
T =
 e f e0
g h g0

b0
d0 

f0 
h0
T will be contracting on L1 ({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 operator T is contracting on L∞ ({1, 2}, R × R) if the
terms of the matrix T satisfy


|e| + |g| + |e0 | + |g 0 | ≤ 1
|a| + |c| + |a0 | + |c0 | ≤ 1








 |a| + |c| + |b0 | + |d0 | ≤ 1
 |e| + |g| + |f 0 | + |h0 | ≤ 1
and


|b| + |d| + |a0 | + |c0 | ≤ 1
|f | + |h| + |e0 | + |g 0 | ≤ 1








|b| + |d| + |b0 | + |d0 | ≤ 1
|f | + |h| + |f 0 | + |h0 | ≤ 1
Let T be the linear operator on L1 ({1, 2}, R × R) represented by a square matrix
of order 4 defined by


2/9 0 3/8 3/7
 1/4 1/9 2/9 2/9 

T =
 0 1/7 2/7 1/4 
2/5 1/9 2/9 3/10
T is contracting on both L1 ({1, 2}, R × R) and L∞ ({1, 2}, R × R).
Consider also the sequence αn = eiθn where θ a complex number, we have by
Theorem 1.2 the convergence of the sequence

k  
2/9 0 3/8 3/7
x1




1
1
1/4
1/9
2/9
2/9
  y1 
αk T k f =
eiθk 



0 1/7 2/7 1/4
x2 
n+1
n+1
k=0
k=0
2/5 1/9 2/9 3/10
y2
n
X
for all f =
x1
y1
n
X
ϕ{1} +
x2
y2
ϕ{2} ∈ L1 ({1, 2}, R × R).
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
9
2. Let X be a reflexive Banach space, Ω a probability space, ϕ is a transformation from Ω to Ω such that for all f ∈ L1 (X),
Z
f ◦ ϕ(ω) dµ(ω) ≤
Ω
Z
f (ω) dµ(ω)
Ω
(or ϕ is a measure preserving transformation) and let (αi )i∈N be any Besicovich
bounded sequence then
n
1 X
lim
αi (f ◦ ϕi )
n n+1
i=0
exists a.e. for every f ∈ L1 (X).
Applications. The general result of Theorem 1.2 can be applied to give a generalization of the vector valued random ergodic theorem of Beck and Schwartz [1].
Theorem 1.6. Let be defined on Ω a strongly measurable function Uω with values in the Banach space B(X) of bounded linear operators on a space X. Suppose
that kUω k ≤ 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 for f ∈ L1 (X), the
limit
n
1 X
αk Uω Uφ(ω) . . . Uφk−1 (ω) f (ϕk (ω))
lim
n n+1
k=0
exists for almost all ω ∈ Ω.
Proof. For f ∈ L1 (X) we define
U f (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 operators T1 , . . . , Td
which more general means difficult. But if T1 = T s1 , . . . . Td = T sd where T is an
linear operator on L1 (X) and sk ∈ N for k = 1, . . . , d, then Theorem 1.2 can be
extended to this case. Let
Bn (T, d, α, f ) =
n
n
X
X
1
·
·
·
αλ T i1 . . . Tdid f
(n + 1)d i =0 i =0 d 1
1
d
∗
where
αj be a bounded
λd = i1 s1 + . . . id sd and
Besicovich sequence. Let fd =
∗
∗
supn Bn (T, d, α, f )X and ed (a, α) = fd > αa .
10
K. EL BERDAN
Theorem 2.1. Let X be a reflexive Banach space, T be a linear operator on
L1 (X) contracting in L1 (X) and in L∞ (X), αj be a bounded Besicovich sequence.
Then for λd = i1 s1 + · · · + id sd , s = s1 + . . . , sd :
(i) For f ∈ L1 (X), et a > 0 we have
Z
e∗
d (a,α)
a − f a− (ω)X dµ(ω) ≤ s
Z
Ω
a+ f (ω) dµ(ω).
X
(ii) For f ∈ L1 (X), the limit of Bn (T, d, α, f )(ω) exists strongly for every
ω ∈ Ω as n tends to infinity.
(iii) If 1 < p < ∞, f ∈ Lp (X) and α = supk αk , the average Bn (T, d, α, f )
converges a.e. and
1/p
p
kf kp .
sup Bn (T, d, α, f )X ≤ α
p−1
p
n
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 the d-dimensional case (d > 2).
We now study the following averages:
Bn (T, 2, α, f )(ω) =
n
n X
X
1
αi+j T i+j f.
(n + 1)2 i=0 j=0
By the relation (3) in the proof of the Theorem 1.2 we can write
n X
n
X
αi+j T i+j f =
i=0 j=0
n X
n
X
j+k
n X
n
h
i X
X
αj+k T j+k f a− +
T j+k−m dm +
αj+k f j+k .
m=0
j=0 k=0
j=0 k=0
We prove an analogous equality to (∗) as in the proof of Theorem 1.2
(7)
n X
n
X
h
αj+k T
j+k a−
n n n+j
X
X
j=0
+
T
j+k−m
i
dm =
m=0
j=0 k=0
=
f
j+k
X
|
t=0
αt T t f a− +
t
X
m=0
n n+j
X
X
j=0 t=j
|
t
X
αt T t f a− +
T t−m dm
{z
m=0
}
1
j−1
t
X
o
X
T t−m dm −
αt T t f a− +
T t−m dm
{z
1
t=0
m=0
}
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
=
n nn+j
X
X
j=0
−
n+j−t
X
T t αt f a− +
αt+m dm
t=0
j−1
X
m=0
|
−
j−t−1
n+j−t
h
i
X
X
a−
T αt f +
αt+m dm +
αt+m dm
t
m=0
|
−
T t αt f a− +
t=0
j−1
X
=
j−t−1
X
}
n+j
i
Xh n+j−t
X
αt+m dm +
αt+m dm
m=0
t=0 m=j−t
|
{z
}
3
j−1−t
o
X
T t αt f a− +
αt+m dm
t=0
n n+j
X
X
m=j−t
2
m=0
n nn+j
X
X
j=0
{z
j−1−t
o
X
T t αt f a− +
αt+m dm
t=0
=
by (∗)
m=0
t=0
j−1
X
}
2
j−1−t
o
X
a−
T αt f +
αt+m dm
n nn+j
X
X
j=0
{z
t
t=0
=
11
m=0
j−t−1
n+j
o
X
X n+j−t
X
T t αt f a− +
αt+m dm +
Tt
αt+m dm .
m=0
j=0 t=j
t=0
|
m=j−t
{z
3
}
Let
χn (ω) =
n
n X
X
j+k
h
i
X
αj+k T j+k f a− +
T j+k−m dm (ω)
m=0
j=0 k=0
and
fC∗ = supCn (T, α, f )X
n
where
Cn (T, α, f ) =
n X
n
X
1
αj+k T j+k f
(2n + 1)2 j=0
k=0
and e∗C (a, α) = {fC∗ > aα}. Fix ω ∈ e∗C (a, α) there exists n = n(ω) such that
n
n X
X
(2n + 1)2 αa ≤ αj+k T j+k f (ω)
j=0 k=0
X
n
n X
X
≤ χn (ω)X +
αj+k fj+k X .
j=0 k=0
12
K. EL BERDAN
But
n
h X
i
≤ α0 f a− (ω)X +
αk dk (ω)X
χn (ω)
X
k=0
j−t−1
n n+j
n n+j
n+j−t
X
X
X
X
X
t
t X
a−
αt+m dm (ω) +
αt+m dm (ω) .
+
T αt f +
T
X
m=0
j=0 t=j
j=0 t=0
X
m=j−t
We know that T is contracting in L∞ (X), using (1) we obtain:
χn (ω)
X
n
n n+j
n n+j
h X
X
X
X
i X
≤ α0 f a− (ω)X +
αk dk (ω)X +
aα +
aα
j=0 t=0
j=0 t=j
k=0
n
n
n
h X
X
i X
≤ α0 f a− (ω)X +
αk dk (ω)X +
(n − j − j)αa +
(n + j)αa .
j=0
k=0
j=0
By (3) we can write
n
h X
i
αk dk (ω)X
(2n + 1)2 αa ≤ α0 f a− (ω)X +
k=0
+ n(n + 1)αa + 2n2 αa +
n X
n
X
j=0 k=0
αj+k fj+k X
n
n
n X
h
X
X
i
dk (ω) + (3n2 + n)αa + α
fj+k .
≤ α f a− (ω)X +
X
X
j=0 k=0
k=0
As in the case d = 1 we have by the relations (3), (4) and (5)
n
∞ X
∞
h
X
i X
dK (ω) +
fj+k (ω) .
a ≤ f a− (ω)X +
X
X
j=0 k=0
k=0
By the (5) we see for all ω ∈ e∗C (a, α)
∞
X
dk (ω) .
a = f a− (ω)X +
X
k=0
This implies
Z
e∗
C (a,α)
∞
X
dk ≤ f0 =
a − f a− (ω)X dµ(ω) ≤
1
1
k=0
On the other hand we can write
Bn (T, α, f ) =
(2n + 1)2
Cn (T, α, f )
(n + 1)2
Z
Ω
f0 (ω) dµ(ω).
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
13
which gives fB∗ ≤ 2fC∗ hence e∗B (a, α) ⊆ e∗C (a/2, α). For b = a/2 we have
Z
Z
a − f a− (ω)X dµ(ω) ≤ 2
b − f b− (ω)X dµ(ω)
e∗
B (a,α)
e∗
C (a,α)
Z
≤2
Ω
b+ f (ω) dµ(ω) ≤ 2
X
so
(8)
aµ e∗B (a, α) ≤ aµ e∗C (a/2, α) ≤ 2
Z
Z
Ω
a+ f (ω) dµ(ω)
X
f (ω) dµ(ω).
Ω
Now, we shall prove that averages
B(n1 , n2 , T, α, f ) =
n1 X
n2
1 X
ϕ1 (i)ϕ2 (j)T i+j f
n1 n2 i=0 j=0
converge on a dense set in L1 (X). We will need the following lemma:
Lemma 2.2. Let T be a linear operator on L1 (X) which is contracting in both
L (X) and L∞ (X), then for f ∈ Inv (T ) + Im (I − T ) ∩ L∞ (X) the limit
1
lim
n1 X
n2
1 X
T i+j f
n1 n2 i=0 j=0
exists as n1 and n2 tend to infinity.
Proof. Let f = g + (h − T h) with T g = g, g ∈ L1 (X) and h ∈ L∞ (X). Then
n2
n1 X
n2
n1
hX
i
1 X
1 X
T i+j f = g +
Ti
(T j h − T j+1 f )
n1 n2 i=0 j=0
n1 n2 i=0
j=0
=g+
=g+
n1
i
i
1 X
T h − T n2 −1 f
n1 n2 i=0
n1
n1
i
i
1h1 X
1h1 X
T ih −
T n2 −1+i h .
n1 n2 i=0
n1 n2 i=0
But T ∞ ≤ 1 hence
n1
1h1 X
i
1
→∞
h −n−2−
T ih ≤
−→ 0
∞
n2 n1 i=0
n2
∞
and
n1
h1h1 X
i
1
→∞
h −n−2−
T n2 −1+i h ≤
−→ 0.
∞
n2 n1 i=0
n2
∞
14
K. EL BERDAN
Let Uk f = eiθk Tk f , k = 1, 2. Uk is a linear operator satisfying the conditions
of Lemma 2.2 and so the theorem holds in the case φk (n) = eθk n .
P 1 Pn2 1
2
i+j
This implies that n11n2 ni=0
f converges a.e. on Inv (T ) +
j=0 ϕ (i)ϕ (j)T
1
Im (I − T ) ∩ L (X) which is dense, by Kakutani-Yoshida theorem in L1 (X) (X a
reflexive Banach space). From Theorem 2.1(i) and the linearity of convergence of
sequences we obtain that
n1 X
n2
1 X
ϕ1 (i)ϕ2 (j)T i+j f
n1 ,n2 →∞ n1 n2
i=0 j=0
lim
exists and is finite a.e. for any trigonometric polynomial φk , 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 rearrangement formula used in part I.
Let αj = 1, and sk = 1 for all j in N and k = 1, . . . , d the average Bn (T, d, α, f )
becomes
Bn (T, d, α, f ) =
n
n
n
1 X
d
X
X
1
i1 +···+id
·
·
·
T
f
=
T j f.
d
(n + 1) i =0 i =0
n + 1 j=0
1
d
Using Theorem 2.1, we deduce the following:
Corollary 2.3. Let X be a reflexive Banach space, T be a linear operator on
L1 (X) contracting in L1 (X) and in L∞ (X), then for d ∈ N and f ∈ L1 (X)
lim
n
1 X j d
T
f
n + 1 j=0
n
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.
THE VECTOR INDIVIDUAL WEIGHTED ERGODIC THEOREM
15
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, HadethBeirut, (Mazraa P. O. Box: 14 - 6 573), Lebanon; e-mail: kberdan@inco.com.lb