A zero-two theorem for a certain class of positive contractions in finite dimensional <span class="mathjax-formula">$L^P$</span>-spaces <span class="mathjax-formula">$(1 \leqslant p < + \infty )$</span>

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 L

^P

-spaces (1 6 p < +∞)

Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 78, série Probabilités et applications, n

^o

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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A

Zero-Two Theorem for

^a

certain class of positive

Contractions in Finite Dimensional Lp-spaces ^{(1 ~ P}

⁺

^~)

R.

ZAHAROPOL

Summary

Our

goal

^{here is to} extend Theorem 1.1 from

[2] (which

is sometimes called the zero-two law for

positive

contractions in

L-spaces)

^{to a class of}

positive

contractions in finite

dimensional LP-spaces ⁽¹

^{p +}

^{oe) .}

1. A General Lemma

Let

(X,Z,mY

^be a measure space and

LP(X,E,m) ⁽¹

^p

^+~)

^{the usual}

Banach _spaces.

By

^a

positive

contraction we mean that T is a linear bounded

operator

which transforms

non-negative

^{functions into non-}

negative

^{functions and its norm is not more than one.}

Lemma 1. Let

1

^{p + m and let -~- be a}

^positive

contraction.

Suppose

^that there exist c &#x3E; 0 and

nO

^{E N U}

^fOl

^{such that}

I IT no+j _ ^{2(l e) .}

^{Let f E} be such that for every p

Clearly

^{it is}

enough

^to prove the lemma f or

IIfll

_p ^{- 1 .}

1. This

_paper

^is part ^{of the} author’s Ph.D. Thesis

^at

the Hebrew University ^of Jerusalem. I wish to

express my

deepest gratitude ^{to Professor} Harry Furstenbergy my supervisory ^for

his valuable support ^and guidance.

If f is as above then

Using

the fact that

n

is a

decreasing

sequence ^we obtain that p ⁿ

It follows that if ^we note p =

(1 - e) 1/p

^then

and we obtain that

By

^{induction it} follows that for _every hEN

p~

^and

^using

the fact that a

decreasing

sequence it follows that lim

p ⁿ _n p

Remark. In Lemma 1 ^we may

drop

^the

assumption

^{of T}

being positive.

^However

we will need this

assumption

^{later on.}

~

2. The Finite Dimensional

Lp-spaces

and the Theorem We will now consider the

following

^case:

Let

k ~ N , k &#x3E;

2 be and we note X ⁼

{l,2,...,k} ,

^{E =}

^{P(X) .}

^Let

m1 ...,mk

^{be k non-zero}

^positive

^{real numbers. We will denote}

^by

^{m the}

measure

generated by mk ^(that

^is

^{m~{il) -} m,,

^{i -}

^l,...,k).

^{We will}

call the _space a finite dimensional

Lp-space

^{and we will note}

t P (k,m)

⁼

L’(X,E,m) .

^A

positive

contraction

tp(k,m)

^is

^generated

by

^{a matrix}

(a )

^{and the}

resulting positive

contraction

T n (n

^E

^N)

is

generated by

i,j=l,..,.Ik

°

If T is a

positive

contraction ^{on we} will note

o -

I

^{for every n E N}

Lemma 2. Let T:

91(k,m)

^{be a}

^positive contraction. Then

the

following

^are

equivalent:

a)

^for every n E N U

{0}

Proof.

a)~b) Suppose Q=o.

^{It follows that for} every i E

{l,2,...,k}

k

(ni)

there

exists ;

^{E N} such that E

aij

^m

mi

^or there exists

J.

^{such that}

~

J ^{J i}

(ni+1) ⁰

^{0 . In other words} for every i

E {1,2,,...,,kl

there exists

a..

^.

^{aij =}

^{o. In} other words for every i ~

{1,2,...,k}

there exists

0

0

--

ni

^{such that}

If we note it follows ^that for and for

every i E

{l,2,...,k} 11 (T n+l - _2m, ^(as

^{for every i E .}

{l,2,...,k}

It follows that 2 .

b) ~a)

^{If then for} every ⁿ E N U

{O}

^{and i} E 0

=

2))l- JL

^{and as T is a}

^positive

contraction it

follows

that for _every

nENU{0}lln+

Now ^we are able to prove the desired result:

Theorem 3. Let p be such that _{1 p} + - and let T be

simultaneously

^a

positive

contraction of and

tp(k,m) .

^If there exists

n 0

^{E N U}

^{01

JL p ^U

Proof . If then 0

(as

every ^{two norms} in

k2

ⁿ

^"P

ⁿ

ll1 R

2

are

equivalent) . Using

^{the zero-two law for}

positive

contractions in

I.1-spaces ^(Theorem

^{1.1 from}

^[2])

^{it follows that for every n E N U}

^{0}

‘I Tn+1 _ Tn,,

^{2 and}

^by

^Lemma 2 it follows that

9

^{If i E S2 then the}

characteristic function

1{i}

^{satisfies the} conditions of Lemma 1 and ^it follows

that lim We obtain that lim

n n

which contradicts the fact that i

Reference’s

1.

Neveu,

^J.:

"Mathematical

foundations of the calculus of

probability",

^San

Francisco, London,

Amsterdam: Holden

Day 1965.

2.

Ornstein,

^{D. and}

Sucheston,

^{L.: "An}

operator

^{Theorem on L.} convergence to zero with

applications

^{to Markov}

kernels".

^{Ann. Math.}

Statist.

¹⁹⁷⁰

vol.

41,

^no.

5,

^1631-1639.

R. ZAHAROPOL

Département

^de

Mathématiques

Université

Hgbraique

GIVAT-RAM Jerusalem Israel

Requ

^en

Septembre

¹⁹⁸³