On the extreme eigenvalues of block $2\times2$ Hermitian matrices

The lower bound
$$ \lambda_1(A)-\lambda_n(A)\ge2\|A_{12}\| $$
for the difference of the extreme eigenvalues of an $n\times n$ Hermitian block $2\times2$ matrix $A=\left[\smallmatrix A_{11}&A_{12}\\A^*_{12}&A_{22}\endsmallmatrix\right]$ is established, and conditions necessary and sufficient for this bound to be attained at $A$ are provided. Some corollaries of this result are derived. In particular, for a positive-definite matrix $A$, it is demonstrated that $\lambda_1(A)-\lambda_n(A)=2\|A_{12}\|$ if and only if $A$ is optimally conditioned, and explicit expressions for the extreme eigenvalues of such matrices are obtained.