When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold? Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\left\Vert Ax\right\Vert_2}{\left\Vert x \right\Vert_2}$ where the the norm in the numerator and denominator are the standard Euclidean norm.
If $\mathbf{N}$ and $\mathbf{M}$ are normal matrices on a separable complex Hilbert space $H$, and $f$ is a Lipschitz function defined on the spectrum of both matrices $\Omega = \sigma(\mathbf{N}) \cup  \sigma(\mathbf{M}) $ with Lipschitz constant $k$ then $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_\mathrm{F} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_\mathrm{F}$. This is a result from Kittaneh (1985). Note that in this paper they use the notation $\left\Vert \cdot \right\Vert_2$ to be the Hilbert–Schmidt operator which I believe is the Frobenius norm in the finite dimensional case.
I found this recent survey paper on operator Lipschitz functions which states similar results. From what I can tell the norm isn't specified but I think the results are also for the Frobenius norm.
I would like to know

*

*Is the term "operator Lipschitz function" reserved for functions which have the property under the Frobenius norm, or is it a more general concept defined for any norms?

*Does the result from Kittaneh (1985) hold for operator norms and if so what is the reference? (The proof in this paper seems specific to the Frobenius norm). The review paper says for equation 3.1.2 "It follows easily from the spectral theory for pairs commuting normal operators", I'm not quite sure what this is and if it holds for norms generally (I haven't had much much formal training in functional analysis or measure theory)

Specifically for my research I have real symmetric matrices $\mathbf{N}$ and $\mathbf{M}$ with eigenvalues lying in the interval $[-1, 1]$. If I have a Lipschitz continuous function $f:[-1,1] \rightarrow \mathbb{R}$ (we can also add differentiability or infinitely differentiability as an assumption if it helps) does it hold that $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ where $k$ is the Lipschitz constant of $f$?
I hope this question isn't too basic for MO, I asked on the mathematics SE but didn't get a response.
 A: The term "operator Lipschitz function" is definitely not reserved to the Hilbert-Schmidt norm. On the opposite, I would say that it is mostly used for the operator norm (but not only, see for example https://arxiv.org/abs/0904.4095 ). In particular, the survey that you are citing is using the operator norm.
It is known that not every Lipschitz function is operator Lipschitz (for the operator norm). So the answer to you question 2) is negative. I think that this was a conjecture of Krein, and that the first  countexample was produced by Farforovsakaya (1972). A very simple example (Davies and Kato) is given by the absolute value map, which is $1$-Lipschitz but only $\simeq \log(n)$-Lipschitz on $M_n(\mathbf{C})$ (for the operator norm). This is very closely related to the fact that the triangular truncation has norm $\simeq \log(n)$ on $M_n(\mathbf{C})$. In fact, any $1$-Lipschitz function is $O(\log n)$-Lipschitz on $M_n(\mathbf{C})$.
In another direction, sufficient conditions are known to ensure that a function is operator-Lipschitz (for the operator norm), in terms of Besov spaces (Peller, see the survey you cite). It is also known that Lipschitz functions are $O(\max(p,1/(p-1))$-Lipschitz on $S^p$, the Schatten $p$-class. This is the paper I cite above by Potapov and Sukochev.
Note: I am not sure, but I would expect that the fact that Lipschitz functions remain Lipschitz for the Hilbert-Schmidt norm was already known to Krein in the 1960's.
A: There are several inaccuracies or mistaken impressions in the OP's original question – I say this not to be denigrating, since it is always tricky reading the literature in a different mathematical community from one's own. However it seems worth writing an answer rather than "sniping in comments".

First of all, Aleksandrov and Peller have written many papers on this theme, but in their setting they almost always mean the operator norm not the Frobenius norm. Hence, when you link to their paper https://arxiv.org/abs/1611.01593 you cannot use it to say things about the Kittaneh paper.
Regarding
Q1. My impression is that among the infinite-dimensional functional analysis community, "operator Lipschitz" is interpreted with respect to the operator norm. This is for instance what is meant by the Aleksandrov—Peller paper that you linked to, contrary to your impression.
Q2. Note that the equation in A+P that you refer to is for commuting operators only! When two matrices commute and are normal in the technical sense of that word, they can be simultaneously diagonalized (over the complex numbers) and hence estimates of the form you seek can be obtained directly. The difficulty, in both the Kittaneh paper and the series of papers by A+P, lies in the fact that the given matrices might not commute with each other.
Finally, there is an example going back to Kato, and refined in work of Davies I think, which shows that the absolute value function is not "operator Lipschitz". So in full generality the answer to Q2 is no. (I don't have the references to hand but will try to update this later)
