The spectral measure and Hilbert transform of a measure-preserving transformation

Abstract

V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on L 2 {L^2} sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.