Let $0<\beta<1$ and $ f \colon [0,1] \to [0,1]$ be $\beta$ Hölder continuous with constant $C$. Let $H$ be a Hilbert space and $A,B$ be self adjoint operators on $H$, such that $\sigma(A+B),\sigma(A) \subset [0,1]$. Then we can define $f(A+B)$ and $f(B)$ by the continuous functional calculus. Do we then have the estimate
$$ \left \lvert \operatorname{tr} (f(A+B)-f(A)) \right \rvert \le C \lVert B \rVert\_\beta^\beta$$
EDIT: The semi-norm $\lVert B \rVert\_\beta$ is the Schatten von Neumann semi-norm.


This does hold for commutating operators $A,B$ and it seems to hold for 2x2 matrices, if i calculated correctly. There is also the stronger hypothesis, that for any unitary equivalent norm $\lVert \cdot \rVert$, we have the estimate
$$ \left \lVert f(A+B) - f(A) \right \rVert \le C \lVert \lvert B \rvert^\beta\rVert$$
I am aware of the question [Hölder continuity for operators](https://mathoverflow.net/questions/308048/h%c3%b6lder-continuity-for-operators) and its answer, but this is different, as the trivial counter example does not hold. The special case $f(t)=t^\beta$ is stated as true in an answer to that question.



Hölder continuity of functional calculus
Such questions have been much studied, in particular by Aleksandrov and Peller. Probably the most relevant reference is the paper *Functions of operators under perturbations of class $S\_p$* by Aleksandrov and Peller,
J. Funct. Anal. 258 (2010). [Zbmath link](https://zbmath.org/?q=an%3A1196.47012) or [mathscinet link](https://mathscinet.ams.org/mathscinet-getitem?mr=2606869).


In particular it is proved there (Theorem 9.14) that for every $\beta<1$ and $p \leq 1$, there is a $\beta$-Hölder-continuous function $f$ and operators $A$, $B$ such that $B \in S\_{1}$ such that $f(A)-f(B)$ does not belong to $S\_{1/\beta}$. In particular, $B \in S\_\beta$ and $f(A) -f(B)$ does not belong to $S^1$.


Remarkably, this is optimal (Theorem 9.13): for every $p>1$, and every $\beta$-Hölder continuous function $f$, $f(A+B) - f(A)$ belongs to $S\_{p/\beta}$ whenever $B$ belongs to $S\_p$.


In the same paper, sufficient conditions on $f$ which imply that $\|f(A+B)-f(A)\|\_1 \leq \|B\|\_\beta^\beta$ are derived.


