  It is shown that to every operator T in a general von Neumann factor M of
type II_1 and to every Borel set B in the complex plane, one can associate a
largest, closed, T-invariant subspace, K = K_T(B), affiliated with M, such that
the Brown measure of T|_K is concentrated on B. Moreover, K is
T-hyperinvariant, and the Brown measure of (1-P_K)T|_(1-P_K)(H) is concentrated
on C\B. In particular, if T has a Brown measure which is not concentrated on a
singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace.
Furthermore, it is shown that for every T in M, the limit
A=\lim_{n\to\infty}[(T^n)* T^n]^{1/2n} exists in the strong operator topology
and K_T(\bar{B(0,r)})=1_{[0,r]}(A), r>0.
