Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ applied to the eigenvalues of $A$. That is, if $A = \sum \lambda_i v_iv_i^\dagger$ is a unitary eigendecomposition of $A$, $f(A) = \sum f(\lambda_i) v_iv_i^\dagger$.

**Question.** Is it true for spectral norm that, for some universal constant $C$, $$ \| f(A) - f(B) \| \overset{?}{\leq} CL\|A - B\| $$

It is true for Frobenius norm [Cor 2.3, Gil] and for spectral norm with an additional $\log^2$ term [Farforovskaya, Nikolskaya]: $$ \|f(A) - f(B)\| \leq 4L\|A-B\|\Big[\log\Big(\frac{s}{\|A-B\|}+1\Big)+1\Big]^2,$$ where $s := \max\{\lambda_i,\mu_i\} - \min\{\lambda_i,\mu_i\}$ is the size of the spectra. I don't know if this is state-of-the-art or whether the log terms are inherent somehow.

I've looked through some relevant matrix analysis textbooks (Horn & Johnson - Topics in Matrix Analysis, Higham - Functions of Matrices, Theory and Computation, Bhatia - Matrix Analysis), which seem useful because when $f$ is differentiable this can be solved by bounding the spectral norm of a Fréchet derivative. But I only found things that were close but mostly for Frobenius norm. [Topics in Matrix Analysis p559 Problem 43] basically gives the result for commuting $A,B$, but this restriction is too strong.

References:

- Michael I. Gil. Perturbations of functions of diagonalizable matrices.
*Electronic Journal of Linear Algebra*, 20, 2010. - Yu. B. Farforovskaya and L. Nikolskaya. Modulus of continuity of operator functions.
*St. Petersburg Mathematical Journal*, 20(3):493-506, 2009.