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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.

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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 accompagnying my book Matrices, GTM 216 (Springer-Verlag).

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    $\begingroup$ "The norm you consider is usually called ..." or the spectral norm, or the norm induced by the $\ell^2$ norm, or... $\endgroup$ – Mark Meckes Dec 4 '12 at 14:57
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    $\begingroup$ And of course strictly speaking the OP's definition of that norm is only correct for Hermitian matrices. $\endgroup$ – Mark Meckes Dec 4 '12 at 14:58

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