Estimating Norms of Commutators

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 ‖[x^1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ η_f(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.

2000 AMS Subject Classification::

• 47B47
• 47A56
• 47A58

Keywords:

• commutators
• matrix function
• functional calculus
• normal operator
• spectral norm
• Monte Carlo methods

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.