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 , Joanna 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