A NNALES
SCIENTIFIQUES
DE L’U NIVERSITÉ DE
C LERMONT-F ERRAND 2
Série Probabilités et applications
R. Z AHAROPOL
A zero-two theorem for a certain class of positive contractions
in finite dimensional LP -spaces (1 6 p < +∞)
Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 78, série Probabilités
et applications, no 2 (1984), p. 9-13
<http://www.numdam.org/item?id=ASCFPA_1984__78_2_9_0>
© Université de Clermont-Ferrand 2, 1984, tous droits réservés.
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9
A Zero-Two Theorem for
a
certain class of
Lp-spaces
Contractions in Finite Dimensional
positive
(1 ~
~)
+
P
R. ZAHAROPOL
Summary
Our
here is to extend Theorem 1.1 from
goal
zero-two law
for positive contractions in
contractions in finite
1.
L-spaces)
LP-spaces
(1
is sometimes called the
to a class of
+
p
positive
oe) .
A General Lemma
(X,Z,mY
Let
Banach spaces.
T
dimensional
[2] (which
is
be
By
a
a measure
space and
LP(X,E,m)
negative functions and its
Lemma 1.
Let
1
norm
is not
p
Clearly
2(l e) .
it is
than
and let
+ m
Let
enough
&#x3E;
0
one.
be
and
nO
a
E N U fOl
positive
such that
be such that for every
f E
to prove
non-negative functions into
-~-
c
the usual
we mean
more
Suppose that there exist
contraction.
I IT no+j _
p
+~)
p
positive contraction
linear bounded operator which transforms
a
(1
the lemma f or
IIfll p
-
1 .
1. This paper is part of the author’s Ph.D. Thesis at the
University of Jerusalem. I wish to express my deepest
Hebrew
gratitude
to Professor
his valuable
Harry Furstenbergy
support and guidance.
my
supervisory
for
that
non-
10
If
f
is
above then
as
Using the fact that
It follows that if
and
we
is
we note
a
n
p
p =
(1 -
decreasing
e) 1/p
obtain that
we
then
obtain that
By induction it follows that for every
the fact that
a
p
In Lemma 1
Remark.
we
sequence
we
n
decreasing
p~
hEN
sequence it follows that
and using
lim
p
n
drop
may
the
will need this assumption later
assumption of
T
However
being positive.
on.
~
2.
The Finite Dimensional
We will
k ~ N ,
Let
m1 ...,mk
measure
now
be
k
Lp-spaces
consider the following
k &#x3E;
2
be and
non-zero
a
case:
we note
positive
generated by mk
call the space
and the Theorem
X
=
{l,2,...,k} ,
real numbers.
(that
is
=
P(X) .
We will denote
m~{il) - m,,
finite dimensional
E
i -
Lp-space
by
l,...,k).
and
we
Let
m
the
We will
will note
11
t P (k,m) L’(X,E,m) .
A
=
matrix
by
a
is
generated by
positive contraction
(a )
and the
tp(k,m)
is
generated
T n (n
resulting positive contraction
E N)
°
i,j=l,..,.Ik
If
T
is
I
o -
Lemma 2.
Let
following
are
a)
ni
we
will note
be
91(k,m)
a
positive contraction. Then
the
equivalent:
n E N U
exists ;~
.
on
n E N
{0}
It follows that for every
a)~b) Suppose Q=o.
Proof.
a.. 0
for every
T:
for every
there
positive contraction
a
(ni+1)
=0
0
aij
k
E
such that
E N
J
0 .
o.
i E
{l,2,...,k}
(ni)
aijJ
m
i
or
mi
In other words for every
i ~
E
there exists
{1,2,,...,,kl
{1,2,...,k}
such that
J.
there exists
--
such that
If
and for
it follows that for
we note
.
every
i E
{l,2,...,k}
11 (T n+l -
2m,
(as for
every
i E
{l,2,...,k}
12
the sequence
decreasing one).
°
2 .
It follows that
b) ~a)
=
then for every
If
and
2))l- JL
as
is
T
a
E N U
n
positive
{O}
and
i E 0
contraction it follows that for every
nENU{0}lln+
Now
we are
Theorem 3.
Let
able to prove the desired result:
p
be such that
positive contraction of
k22
R
are
I.1-spaces
(Theorem 1.1 from [2])
‘I Tn+1 _ Tn,,
2
and
by
characteristic function
that
tp(k,m)
p
.
zero-two law
ll1
for
simultaneously
0
(as
n U0
E N U
every two
9
n
If
a
{01
norms
in
E N U {0}
i E S2
then the
satisfies the conditions of Lemma 1 and it follows
We obtain that
lim
lim
n
n
which contradicts the fact that
be
positive contractions
it follows that for every
Lemma 2 it follows that
1{i}
T
If there exists
n
Using the
equivalent) .
and let
then
"P
n
+ -
p
and
JL
If
Proof .
1
i
in
13
Reference’s
"Mathematical foundations of the calculus of probability", San
Francisco, London, Amsterdam: Holden Day 1965.
1.
Neveu, J.:
2.
Ornstein,
to zero
vol.
41,
"An operator Theorem on
D. and Sucheston, L.:
with applications to Markov kernels". Ann. Math.
no.
L.
convergence
1970
Statist.
5, 1631-1639.
R.
ZAHAROPOL
Département
Université
de
Mathématiques
Hgbraique
GIVAT-RAM
Jerusalem
Israel
Requ
en
Septembre 1983