  Results of Haagerup and Schultz (2009) about existence of invariant subspaces
that decompose the Brown measure are extended to a large class of unbounded
operators affiliated to a tracial von Neumann algebra. These subspaces are used
to decompose an arbitrary operator in this class into the sum of a normal
operator and a spectrally negligible operator. This latter result is used to
prove that, on a bimodule over a tracial von Neumann algebra that is closed
with respect to logarithmic submajorization, every trace is spectral, in the
sense that the trace value on an operator depends only on the Brown measure of
the operator.
