Multivariable pseudospectrum in C⁎-algebras

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. In particular, we focus on the forms of multivariable pseudospectra that are finding applications in physics. 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. We prove a relation between the quadratic pseudospectrum and Clifford pseudospectra, as well as results about how symmetries in a tuple of operators can lead to a symmetry in the various pseudospectra.