For operators $S, T\in B(H)$, if $\|S - T\|$ is small, then $\||S| - |T|\|$ is small I know that if $\|S - T\|$ is small, then by two applications of the triangle inequality we get
$$
\|S^*S - T^*T\| \le (\|S\| + \|T\|)\|S - T\|
$$
so that $\||S|^2 - |T|^2\|$ is small. Is it also the case that $\||S| - |T|\|$ is small? I'm fairly certain the square root function is "operator Lipschitz", which would imply the result, but I imagine there's a simpler method which is eluding me. I also know $\||S| - i|T|\| = \|S^*S - T^*T\|^{1/2}$ (when $S$, $T$ commute, thanks to MartinArgerami for pointing out the mistake), but it's not quite what I'm looking for. What am I missing?
Edit: I've found a relevant result from Davidson's "C*-algebras by example", Exercise II.8. If $f : X\to\mathbb{C}$ is a continuous function on a compact set $X$, then for all $\epsilon > 0$ there exists $\delta > 0$ such that whenever $A, B$ are normal operators with $\sigma(A)\cap\sigma(B)\subseteq X$, $\|A - B\| < \delta$ implies $\|f(A) - f(B)\| < \epsilon$. This gives a positive solution whenever $S, T$ are such that $\sigma(S^*S)\cup\sigma(T^*T)\subseteq [\varepsilon, K]$ for arbitrarily small $\varepsilon$ and arbitrarily large $K$.
 A: As stated in the edit to the question, for continuous $f$ the map $A\mapsto f(A)$ is norm continuous on bounded subsets of of the normal operators on $H$. The proof is not hard. It is certainly true if $f$ is a polynomial, and in the general case one can approximate $f$ by polynomials and use the usual continuity properties of functional calculus as a map $f\mapsto f(A)$.
The absolute value is not operator Lipschitz. More precisely, on every infinite-dimensional Hilbert space $H$ there exist sequences $(A_n)$, $(B_n)$ of self-adjoint operators such that $\lVert\lvert A_n\rvert-\lvert B_n\rvert\rVert\geq n\lVert A_n-B_n\rVert$.
However, there are still some quantitative bounds. For example,
$$
\lVert \lvert A\rvert-\lvert B\rvert\rVert\leq \frac 2\pi\lVert A-B\rVert\left(2+\log\frac{\lVert A\rVert+\rVert B\rVert}{\lVert A-B\rVert}\right).
$$
Both of these facts have been proven in T. Kato. Continuity of the Map $S\to\lvert S\rvert$ for Linear Operators. The article is short and quite accessible in my opinion.
