# How to prove Halmos’s Inequality

How to prove Halmos’s Inequality?

If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$

I found it from http://www.staff.vu.edu.au/rgmia/monographs/bullen/Dict-Ineq-Supp-Comb.pdf at page 18.

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Suppose $\|I-(AB-BA)\| \lt 1$. Then $(AB-BA)$ is invertible and as (without loss of generality) $A$ commutes with $(AB-BA)$ by hypothesis, we have that $A$ commutes with $(AB-BA)^{-1}$. Therefore $$I = (AB-BA)(AB-BA)^{-1} = A[B(AB-BA)^{-1}] - [B(AB-BA)^{-1}]A$$ and thus $I$ is exhibited as a commutator, and this is well-known to be impossible by a theorem of Wielandt and Wintner; see e.g. Qiaochu's question here for a proof of that fact. Therefore $\|I-(AB-BA)\| \geq 1$.