Abstract
We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) 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.
Notes
1Code for creating the tables in this paper is available at the Lobo Vault, hosted by the University of New Mexico, http://hdl.handle.net/1928/23461.