Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

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 behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

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

“…The projections also behave well with respect to compressions and similarities (see Theorems 2.4.4 and 12.3 in [4]).…”