Projection to Invariant Functions:

[l'Hôpital]

Context:
$T : X \rightarrow X$ is a measure preserving ergodic transformation of a probability measure space $X$. Let $V_n = \{ g | g \circ T^n = g \}$ and $E = span [ \{g | g \circ T = \lambda g,$ for some $\lambda \} ]$ be the span of the eigenfunctions of the induced operator $T : L^2 \rightarrow L^2$, $Tf = f \circ T$.

Problem:
I was reading this paper by Fursternberg and Weiss where they implicitly claim if $f \perp E$ then $f \perp V_n$. However, I don't see how this isso.

Some help would be greatly appreciated. : )

 

[Office_Shredder]

I think you can do an inductive argument. For example f us perpendicular to V₁ obviously. If g is in V2, then g+gT is in E, as is g-gT. So both of these are perpendicular to f, and therefore their sum is as well. You can probably keep working your way up.

 

[l'Hôpital]

Ooh! I like it! Awesome, thanks!

One more question:

They also makes a claim as follows. Let $P : L^2 \rightarrow V_{n}$ be the projection operator. Then, it can be represented as an integral operator with kernel $K(x,y) = l\sum_{i=1}^{l} 1_{A_i} (x) 1_{A_i} (y)$ where $\cup A_i = X$ are $T^n$-invariant sets. I don't see how this is even possible, nor where the $l$ comes into play, or how they can have only finitely many. Any ideas with this one?

 

[l'Hôpital]

Nevermind, got it! Thanks anyways!