Estimating Norms of Commutators

TL;DR

This paper derives estimates for the norms of commutators in C*-algebras, providing bounds for functions of operators based on their commutator norms, with both theoretical and numerical insights.

Contribution

It introduces new bounds for commutator norms involving functions of operators, extending previous results and including numerical evidence for optimal bounds.

Findings

Derived bounds for [f(x), y] in terms of [x, y]

Analyzed specific cases like [x^2, y] and [x^{1/2}, y]

Numerical experiments suggest optimal bounds in many cases

Abstract

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of \delta = \|[x; y]\| we can rigorous calculate the best possible upper bound \|[f(x), y]\| for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.