Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson¹ ; T. A. Gillespie²

¹ Department of Mathematics University of Illinois 1409 W. Green St. Urbana, IL 61801, U.S.A.
² Department of Mathematics and Statistics University of Edinburgh James Clerk Maxwell Building Edinburgh EH9 3JZ, Scotland

Studia Mathematica, Tome 144 (2001) no. 1, pp. 39-61

Résumé

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.

Détail

DOI : 10.4064/sm144-1-2

Keywords: trigonometrically well bounded operator banach space mathfrak denote frak infty sequence weighted discrete ergodic averages frak sum leq right sequence frak infty weighted ergodic averages converges strong operator topology idempotent operator whose range frak whose null space closure i u frak result expands scope traditional ergodic theorem thereby serves link between banach space spectral theory ergodic operator theory develop characterization trigonometrically well bounded operators their ability transfer discrete hilbert transform banach space setting via weighting hilbert averages these results together those weighted ergodic averages furnish explicit expression spectral decomposition trigonometrically well bounded operator banach space terms strong limits appropriate averages powers treat special circumstances where corresponding results obtained weights removed