On Spectral Operators in Finite von Neumann Algebras

An operator on a Hilbert space is said to be spectral if it has a suitably well-behaved `idempotent-valued' spectral measure. Dunford introduced these operators and also provided the following characterization: An operator is spectral iff it is similar to the sum of a normal operator and a quasinilpotent operator that commute with each other. Operators in a von Neumann algebra with a normal, faithful, tracial state have an associated spectral measure (called the Brown measure) and invariant projections (the Haagerup-Schultz projections), which behave well with respect to the Brown measure. In this paper, we study the angles between the Haagerup-Schultz projections for such operators. We show that an operator in a finite von Neumann algebra is similar to the sum of a normal operator and a commuting s.o.t.-quasinilpotent operator iff the angles between its Haagerup-Schultz projections are uniformly bounded away from zero. This lets us provide a new characterization of spectral operators in finite von Neumann algebras. We then estimate the angles between the Haagerup-Schultz projections for a class of operators from free probability called the DT-operators. These involve new estimates on the norms of algebra-valued circular operators. We then show, subject to some mild regularity conditions on the Brown measure of a DT-operator, that they fail to be spectral. This provides a large class of non-spectral but decomposable operators in a finite von Neumann algebra