Some non-spectral DT-operators in finite von Neumann algebras

TL;DR

This paper demonstrates that certain DT-operators with radially symmetric Brown measure are not spectral, using projections and new estimates on algebra-valued circular operators within finite von Neumann algebras.

Contribution

It introduces novel estimates on norms and traces of algebra-valued circular operators and shows non-spectrality of specific DT-operators based on their Brown measure.

Findings

DT-operators with symmetric Brown measure are not spectral

Angles between certain projections are zero

New bounds on norms and traces of algebra-valued circular operators

Abstract

Given a DT-operator $Z$ whose Brown measure is radially symmetric and has a certain concentration property, it is shown that $Z$ is not spectral in the sense of Dunford. This is accomplished by showing that the angles between certain complementary Haagerup-Schultz projections of $Z$ are zero. New estimates on norms and traces of powers of algebra-valued circular operators over commutative C$^*$-algebras are also proved.