Spectral decompositions, ergodic averages, and the Hilbert transform

Abstract

Let U be a trigonometrically well-bounded operator on a Banach space X, and denote byfAn(U)g 1=1 the sequence of (C; 2) weighted discrete ergodic averages of U, that is, An(U) = 1 n X 0<jkj n 1 jkj n + 1 U k : We show that this sequence fAn(U)g 1=1 of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range isfx2 X : Ux = xg; and whose null space is the closure of (I U)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 operatorU 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.