In 'Topics in Random Matrix Theory' by Terence Tao, Exercise 2.4.3 and Exercise 2.4.4 deals with the stability of the empirical spectral distribution (ESD) w.r.t. some perturbations:
Let $M_{n}$ and $N_{n}$ be sequences of random Hermitian matrix ensembles. Let $\mu$ a (deterministic) probability measure.
Exercise 2.4.3. If $\mu_{\frac{1}{\sqrt{n}} M_{n}} \rightarrow \mu$ a.s. and $\frac{1}{n^2} \| N_{n} \|_{F}^2 \rightarrow 0$ a.s., then $\mu_{ \frac{1}{\sqrt{n}} ( M_{n} + N_{n})} \rightarrow \mu$ a.s..
Exercise 2.4.4. If $\mu_{\frac{1}{\sqrt{n}} M_{n}} \rightarrow \mu$ a.s. and $\frac{1}{n} \operatorname{rank} (N_{n}) \rightarrow 0$ a.s., then $\mu_{ \frac{1}{\sqrt{n}} ( M_{n} + N_{n})} \rightarrow \mu$ a.s..
I solved Exercise 2.4.3., which was just a corollary of Lemma 2.4.3. The hint for Exercise 2.4.4 was to use the Weyl inequalities instead of the Wielandt-Hoffman inequality. But Weyl inequalities contain terms of 'operator norm', not 'rank'. Some identities I recall are $$ \| A \|_{op} \le \| A \|_{F} \le \sqrt{\operatorname{rank} (A)} \| A\|_{op}$$ for Hermitian matrix $A$, and $$ \| A \|_{op} = \sigma(A)$$ for diagonalizable matrix $A$, where $\sigma(A)$ is the spectral radius of $A$.
My questions are:
(1) Any relationship between rank and operator norm of a Hermitian matrix?
(2) Does the assumption $\frac{1}{n} \operatorname{rank}(N_{n}) \rightarrow 0$ a.s. implies $\|M_{n} \|_{op} = o(n)$?