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 {α_k}ⁿ_k=l and {β_k}ⁿ_k=1 respectively (counting multiplicity), then, ||A-B||≥min_σεSn||diag (α_k-β_σ(k))|| 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.