Counterexample to a Question on Commutators

Abstract

We show that it is possible for two selfadjoint operators A and B in a Hilbert space H with bounded commutator to have the property that <!-- MATH $\left| A \right|B - B\left| A \right|$ --> is unbounded (where <!-- MATH $\left| A \right|$ --> denotes the positive square root of ). The proof reduces to showing that for all natural numbers n, there exist a bounded positive operator U and a bounded operator V satisfying <!-- MATH $\left\| {UV - VU} \right\| \geqq n\left\| {UV + VU} \right\|$ --> .