  In this article, we obtain a version of the noncommutative Banach Principle
suitable to prove Wiener-Wintner type results for weights in W1-space. This is
used to obtain noncommutative Wiener-Wintner type ergodic theorems for various
types of weights for certain types of positive Dunford-Schwartz operators. We
also study the b.a.u. (a.u.) convergence of some subsequential averages and
moving averages of such operators
