Weakly wandering vectors and weakly independent partitions

Abstract

We first characterize continuous spectrum and purely discrete spectrum of an isometry U of a Hilbert space geometrically by the existence of a spanning system, resp. by the absence, of vectors with infinitely many orthogonal images under powers of U. We then characterize weak mixing and discrete spectrum of an invertible measure preserving transformation of a probability space in terms of the null sets of the space. Finally for two-fold weakly mixing transformations the result on isometries is strengthened by proving the density of the set of partitions with infinitely many mutually independent images in the set of all finite partitions.