Spectral decompositions, ergodic averages, and the Hilbert transform
Tom 144 / 2001
Streszczenie
Let $U$ be a trigonometrically well-bounded operator on a Banach space $\mathfrak X$, and denote by $\{ {{\frak A}}_{n}( U) \} _{n=1}^{\infty }$ the sequence of $( C,2) $ weighted discrete ergodic averages of $U$, that is, $$ {{\frak A}}_{n}( U) ={1\over n}\sum _{0< | k| \leq n}\left ( 1-{| k| \over n+1} \right) U^{k}. $$ We show that this sequence $\{ {{\frak A}}_{n}( U) \} _{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is $\{ x\in {{\frak X}}:Ux=x\} ,$ and whose null space is the closure of $( I-U) {{\frak X}}$. This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and ergodic operator theory. We also develop a characterization of trigonometrically well-bounded operators by their ability to “transfer” the discrete Hilbert transform to the Banach space setting via $(C,1)$ weighting of Hilbert averages, and these results together with those on weighted ergodic averages furnish an explicit expression for the spectral decomposition of a trigonometrically well-bounded operator $U$ on a Banach space in terms of strong limits of appropriate averages of the powers of $U$. We also treat the special circumstances where corresponding results can be obtained with the $(C,1)$ and $(C,2)$ weights removed.
