Angles between Haagerup–Schultz projections and spectrality of operators

Abstract

We investigate angles between Haagerup–Schultz projections of operators belonging to finite von Neumann algebras, in connection with a property analogous to Dunford's notion of spectrality of operators. In particular, we show that an operator is similar to the sum of a normal and an s.o.t.-quasinilpotent operator that commute if and only if the angles between its Haagerup–Schultz projections are uniformly bounded away from zero (and we call this the uniformly non-zero angles property). Moreover, we show that spectrality is equivalent to this uniformly non-zero angles property plus decomposability. Finally, using this characterization, we construct an easy example of an operator which is decomposable but not spectral, and we show that Voiculescu's circular operator is not spectral (nor is any of the circular free Poisson operators).