Operator $\theta$-H\"{o}lder functions with respect to $\left\|\cdot\right\|_p$, $0< p\le \infty$

Abstract

Let $\theta \in(0,1)$ and $(\mathcal{M},\tau)$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (denoted by $S_{d,\theta}$), showing that there exists a constant $d>0 $ depending on $p$, $0<p\le \infty$, only such that every function $f:\mathbb{R}\rightarrow \mathbb{C} \in S_{d,\theta}$ is operator $\theta$-H\"older with respect to $\left\|\cdot \right\|_p$, that is, there exists a constant $C_{p,f}$ depending on $p$ and $f$ only such that the estimate $$\left\|f(A) -f(B)\right\|_p \le C_{p,f}\left\| \left| A-B \right|^\theta \right \|_p $$ holds for arbitrary self-adjoint $\tau$-measurable operators $A$ and $ B$. In particular, we obtain a sharp condition such that a function $f$ is operator $\theta$-H\"older with respect to all quasi-norms $\left\|\cdot \right\|_p$, $0<p\le \infty$, which complements the results on the case for $ \frac1\theta < p<\infty $ by Aleksandrov and Peller, and the case when $p=\infty$ treated by Aleksandrov and Peller, and by Nikol$^\prime$skaya and Farforovskaya. As an application, we show that this class of functions is operator $\theta$-H\"older with respect to a wide class of symmetrically quasi-normed operator spaces affiliated with $\mathcal{M}$, which unifies the results on specific functions due to Birman, Koplienko and Solomjak, Bhatia, Ando, and Ricard with significant extension. In addition, when $\theta>1$, we obtain a reverse of the Birman-Koplienko-Solomjak inequality, which extends a couple of existing results on fractional powers $t\mapsto t^\theta$ by Ando et al.