Distance between normal operators

Abstract:

Lidskii and Wielandt have proved independently that if $A$ and $B$ are selfadjoint operators on an $n$-dimensional space $H$, with eigenvalues $\{ {\alpha _k}\} _{k = 1}^n$ and $\{ {\beta _k}\} _{k = 1}^n$ respectively (counting multiplicity), then, \[ \left \| {A - B} \right \| \geqslant \min \limits _{\sigma \in {S_n}} \left \| {{\text {diag}}\left ( {{\alpha _k} - {\beta _{\sigma (k)}}} \right )} \right \|\] for any unitarily invariant norm on $L(H)$. In this note an example is given to show that this result is no longer true if $A$ and $B$ are only required to be normal (even unitary). It is also shown that the above inequality holds in the operator norm, if $A$ is selfadjoint and $B$ is skew-self-adjoint.