  We study Schur-type upper triangular forms for elements, T, of von Neumann
algebras equipped with faithful, normal, tracial states. These were introduced
in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz
projections. We investigate when the s.o.t.-quasinilpotent part of this
decomposition of T is actually quasinilpotent. We prove implications involving
decomposability and strong decomposability of T. We show this is related to
norm convergence properties of the sequence |T^n|^{1/n} which, by a result of
Haagerup and Schultz, is known to converge in strong operator topology. We
introduce a Borel decomposability, which is a property appropriate for elements
of finite von Neumann algebras, and show that the circular operator is Borel
decomposable. We also prove the existence of a thin-spectrum
s.o.t.-quasinilpotent operator in the hyperfinite II_1-factor.
