# Simultaneous upper triangular forms for commuting operators in a finite von Neumann algebra

## TL;DR

This paper studies the joint spectral measures and projections for commuting operators in finite von Neumann algebras, establishing their spectral support relations and a simultaneous upper triangularization result.

## Contribution

It introduces a simultaneous upper triangularization for finite commuting tuples and analyzes the properties of joint Brown measure and Haagerup--Schultz projections in this context.

## Key findings

- Support of joint Brown measure is within the Taylor joint spectrum
- Joint Brown measure is contained in the left Harte spectrum
- Existence of simultaneous upper triangularization for commuting tuples

## Abstract

The joint Brown measure and joint Haagerup--Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup--Schultz projections are shown to be have well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.