Geodesic
Bounds for the extreme eigenvalues of block $2\times2$ Hermitian matrices
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IX, Tome 301 (2003), pp. 172-194 
L. Yu. Kolotilina. Bounds for the extreme eigenvalues of block $2\times2$ Hermitian matrices. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IX, Tome 301 (2003), pp. 172-194. http://geodesic.mathdoc.fr/item/ZNSL_2003_301_a4/
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     author = {L. Yu. Kolotilina},
     title = {Bounds for the extreme eigenvalues of block~$2\times2$ {Hermitian} matrices},
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     year = {2003},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_301_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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