Bounds for the extreme eigenvalues of block $2\times2$ Hermitian matrices

Let an $n\times n$ Hermitian matrix $A$ be presented in block $2\times2$ form as $A=\left[\begin{smallmatrix}A_{11}&A_{12}\\A^*_{12}&A_{22}\end{smallmatrix}\right]$, where $A_{12}\ne0$, and assume that the diagonal blocks $A_{11}$ and $A_{22}$ are positive definite. Under these assumptions, it is proved that the extreme eigenvalues of $A$ satisfy the bounds
$$ \lambda_1(A)\ge\|A_{12}\|(\|R\|^{-1}+1),\quad |\lambda_n(A)|\le\|A_{12}\|\,\bigl|\,\|R\|^{-1}-1\bigr|, $$
where $R=A^{-1/2}_{11}A_{12}A^{-1/2}_{22}$ and $\|\cdot\|$ is the spectral norm. Further, in the positive-definite case, several equivalent conditions necessary and sufficient for both of the above bounds to be attained are provided.