## Faculty and Staff Publications

#### Title

Estimating norms of commutators

Article

12-2-2013

#### 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 $\|[x^2,y]\|$ and $\|[x^{1/2},y]\|$ for $0\leq x \leq 1$. For larger values of $\delta = \|[x,y]\|$ we can rigorous calculate the best possible upper bound $\|[f(x),y]\| \leq \eta_f(\delta)$ 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.

English

#### Keywords

Commutators, matrix function, functional calculus, normal operator, spectral norm, Monte Carlo methods