Commutator inequalities associated with the polar decomposition

Kittaneh, Fuad

doi:10.1090/S0002-9939-01-06197-4

Commutator inequalities associated with the polar decomposition
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Proc. Amer. Math. Soc. 130 (2002), 1279-1283 Request permission

Abstract:

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

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