Commutator inequalities associated with the polar decomposition

Abstract

Let A = U P A=UP be a polar decomposition of an n × n n\times n complex matrix A A . Then for every unitarily invariant norm | | | ⋅ | | | |||\cdot ||| , it is shown that \[ | | | | U P − P U | 2 | | | ≤ | | | A ∗ A − A A ∗ | | | ≤ ‖ U P + P U ‖ | | | U P − P U | | | , |||\, |UP-PU|^2||| \le |||A^*A-AA^*|||\le \|UP+PU\|\,|||UP-PU|||, \] where ‖ ⋅ ‖ \|\cdot \| denotes the operator norm. This is a quantitative version of the well-known result that A A is normal if and only if U P = P U UP=PU . Related inequalities involving self-commutators are also obtained.