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

TL;DR

This paper explores operator $ heta$-H"older functions within semifinite von Neumann algebras, establishing sharp conditions, extending previous results, and unifying various classes of functions with applications to inequalities and operator spaces.

Contribution

It introduces new operator $ heta$-H"older function classes, provides sharp conditions for their properties across all quasi-norms, and extends existing inequalities in operator theory.

Findings

Existence of a universal constant $d$ for Sobolev-based function spaces.

Characterization of functions that are operator $ heta$-H"older across all $ orm{ullet}_p$ quasi-norms.

Extension of inequalities to fractional powers and unification of results across different operator spaces.

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.