Multivariable pseudospectrum in $C^*$-algebras

TL;DR

This paper explores the theoretical properties of pseudospectra for noncommuting operator tuples in $C^*$-algebras, with detailed examples including projections, position-momentum pairs, and tridiagonal triples.

Contribution

It introduces and analyzes various forms of pseudospectrum for noncommuting Hermitian operator tuples in $C^*$-algebras, providing explicit calculations for complex examples.

Findings

Computed pseudospectra for noncommuting operator pairs and triples

Analyzed universal projections and quantum operators in $C^*$-algebras

Provided insights into the structure of pseudospectra in infinite-dimensional settings

Abstract

We look at various forms of spectrum and associated pseudospectrum that can be defined for noncommuting $d$-tuples of Hermitian elements of a $C^*$-algebra. The emphasis is on theoretical calculations of examples, in particular for noncommuting pairs and triple of operators on infinite dimensional Hilbert space. In particular, we look at the universal pair of projections in a $C^*$-algebra, the usual position and momentum operators, and triples of tridiagonal operators.