From: bruck@math.usc.edu (Ronald Bruck)
Newsgroups: sci.math.research
Subject: Re: Functional Analysis question
Date: 6 Oct 1998 01:00:01 -0500

In article <p8rb3fndwnf5@forum.swarthmore.edu>,
Joanna  <Galois@student.be> wrote:

:I have had a lot of problems with the following proposition:
:"Let V an operator in a Hilbert space H,such that ||V^n||<-C for all
:n.            Prove that        _
:                          1/k * \  V^n(f)---->Pf   when n tends to 
:                                /_
:infinite,where p is not necessarily an orthogonal projection over 
:   { f: V(f)=f }.  
: ( This result is proposed as an exercise on the book:"Functional
:Analysis"",by Reed Simons.)
:   If someone has an idea of how to prove it or know any reference
:about it, please let me know it.I have many weeks trying to solve it
:but it have been very hard for me.  
:                               Thanks in advance

It's known as the Mean Ergodic Theorem.  There is a beautiful little book
by Lee Lorch which contains an elegant derivation of it; as I recall--
because I lent it to a student and never got it back :-(  You should find
a proof in just about ANY functional analysis text.

BTW, I assume your text is Reed/Simon, BTW.  Simons is a different person
altogether ;-)

There is also a nonlinear version of the theorem, but not for mappings V
which satisfy ||V^n|| <= const; it's known that if the constant is too
large, such mappings need not have fixed-points at all.  The "correct"
generalization is to nonlinear V which have Lipschitz constant one or
less, and which map a bounded closed convex set back into itself; in
that case, (x + Vx + ... + V^{n-1}x)/n converges WEAKLY to a fixed-point
of V.

--Ron Bruck