# Angles between Haagerup--Schultz projections and spectrality of operators

## TL;DR

This paper explores the relationship between angles of Haagerup--Schultz projections and spectral properties of operators in finite von Neumann algebras, providing new characterizations and examples related to spectrality and decomposability.

## Contribution

It introduces the uniformly nonzero angles property and links it to spectrality and decomposability, offering new insights and examples in operator theory.

## Key findings

- Operators can be decomposed into normal and quasinilpotent parts if angles are bounded away from zero.
- Spectrality is characterized by the combination of the nonzero angles property and decomposability.
- Voiculescu's circular operator is shown not to be spectral.

## 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 can be written as 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 nonzero anlges property). Moreover, we show that spectrality is equivalent to this uniformly nonzero 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 are any of the circular free Poisson operators).