Let $A\in\mathbb{C}^{n\times n}$ be positive definite Hermitian, where $n\ge 2$, and define $\|X\|_A^2=\mathrm{tr}(AX^*X)$ for all $X\in\mathbb{C}^{n\times n}$. We determine the smallest constant $C_A$ for which $$ \|XY-YX\|_{\mathrm{F}}^2 \le C_A\|X\|_A^2\|Y\|_A^2 $$ holds for all rank-one $X,Y\in\mathbb{C}^{n\times n}$, where $\|\cdot\|_{\mathrm{F}}$ denotes the Frobenius norm. The constant depends only on the two smallest eigenvalues of $A$ and is determined by a strictly monotone scalar equation. This answers the sharp-constant question for rank-one matrices formulated by Fang and Cheng. We also characterize all equality pairs.
Let $A$ and $B$ be unital $C^*$-algebras, and let $\mathrm{S}_{A^+}$ and $\mathrm{S}_{B^+}$ denote their positive unit spheres. We first show that the diametral relation on the positive unit sphere already determines Jordan structure. More precisely, suppose that $\Lambda:\mathrm{S}_{A^+}\to\mathrm{S}_{B^+}$ is a bijec...
Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathca...
We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant r...
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
We study geometric inverse problems for finite sets $A\subset \mathbb{Z}^d$ whose Fourier $L^1$ norm, or Wiener norm, \begin{align*} \|A\|:=\int_{\mathbb{T}^d}\Big|\sum_{a\in A}e(a\cdot t)\Big|dt \end{align*} is subpolynomial in $|A|$. In particular, we show a variety of geometric phenomena are incompatible with small...
Alex Burgin, Jun-Zhe Mao· 0 citations
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