# Spectral norm bound on smooth primary matrix function perturbation

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.