The One-sided Ergodic Hilbert Transform of Normal Contractions

Abstract

Let T be a normal contraction on a Hilbert space H. For f ∈ H we study the one-sided ergodic Hilbert transform $\mathop {\lim }\limits_{n \to \infty } \sum\nolimits_{k = 1}^n {\frac{{T^k f}} {k}}$ . We prove that weak and strong convergence are equivalent, and show that the convergence is equivalent to convergence of the series $\sum\nolimits_{n = 1}^\infty {\frac{{\log n||\sum\nolimits_{k = 1}^n {T^k f||^2 } }} {{n^3 }}}$ . When $H = \overline {(I - T)H}$ , the transform is shown to be precisely minus the infinitesimal generator of the strongly continuous semi-group {(I−T)^r} _r ≥0.

The equivalence of weak and strong convergence of the transform is proved also for T an isometry or the dual of an isometry.

For a general contraction T, we obtain that convergence of the series $\sum\nolimits_{n = 1}^\infty {\frac{{\left\langle {T^n f,f} \right\rangle \log n}} {n}}$ implies strong convergence of $\sum\nolimits_{n = 1}^\infty {\frac{{T^n f}} {n}}$.