Let us suppose that $A\_{n}$ and $B\_n$ are sequences of positive definite matrices satisfying


$$c \leq \lambda\_{\min}(A\_n)\leq \lambda\_{\max}(A\_n)\leq C$$


and


$$c \leq \lambda\_{\min}(B\_n)\leq \lambda\_{\max}(B\_n)\leq C$$


where $\lambda\_{\min}$ and $\lambda\_{\max}$ are minimal and maximal eigenvalues. Then, what's the relationship between $\|A\_n-B\_n\|\_2$ and $\|A\_n^2-B\_n^2\|\_2$? Is it true that 


$$\|A\_n-B\_n\|\_2 \leq \text{constant} \cdot \|A\_n^2-B\_n^2\|\_2$$


where the $2$-norm of a matrix $M$ is the maximum of the absolute value of its minimal eigenvalue and the absolute value of its maximal eigenvalue. 



Relationship between eigenvalues of $A-B$ and eigenvalues of $A^2-B^2$
The norm you consider is usually called *operator norm*, or the norm *subordinated* to the $\ell^2$ norm over ${\mathbb C}^n$ (or ${\mathbb R}\_n$). The correct inequality for positive Hermitian matrices $A$ an $B$ is
$$\|A-B\|\_2\le\sqrt{\|A^2-B^2\|\_2}.$$
See Exercise 110 of my [additional list](http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf) accompagnying my book *Matrices*, GTM **216** (Springer-Verlag).


