Hyponormal Toeplitz Operators on the Bergman Space of the Disk
Nicole Revilla, Brian Simanek
TL;DR
This paper studies hyponormal Toeplitz operators $T_{\varphi}$ on the Bergman space $A^2(\mathbb{D})$ with symbol $\varphi(z)=z^{n}|z|^{2s}+a(t)\bar{z}^{m}|z|^{2t}$ and analyzes how large the perturbation $|a(t)|$ can be for hyponormality as $t\to\infty$ (and also treats $t\to0$). It establishes asymptotic thresholds: if $\limsup_{t\to\infty}|t a(t)|<\frac{n+2s}{2}$ then hyponormality holds for large $t$, while if $\liminf_{t\to\infty}|t a(t)|>\frac{n+2s}{2}$ it fails; it also proves a $t\to0$ bound $|a|^2\le\min\{\frac{(m+1)(n+1)}{(n+s+1)^2},\frac{n(n+2s)}{m^2}\}$. Additionally, the work corrects a mistake in Fleeman–Liaw regarding the norm of the commutator $[T_{z^m\bar{z}^n}^*,T_{z^m\bar{z}^n}]$ and provides sharp bounds for related symbol configurations, with a detailed analysis of a monotonicity condition that yields a lower boundary curve near $m/n\approx1.61$ and a universal bound $\max_k\lambda_k\le\tfrac12$. The authors outline future directions toward a complete Bergman-space symbol classification for hyponormal Toeplitz operators and the precise commutator norm, including the problem of determining, for $m>n$, when $\|[T_{z^m\bar{z}^n}^*,T_{z^m\bar{z}^n}]\|=\lambda_k$ for specific $k$.
Abstract
We consider Toeplitz operators with bounded symbol acting on the Bergman space of the unit disk and assess their hyponormality. We will mainly be concerned with the symbol $\varphi(z)=z^{n}|z|^{2s}+a(t)\bar{z}^{m}|z|^{2t}$, where $s$ and $t$ are positive real numbers and $m$ and $n$ are natural numbers. The main goal is to understand how large $|a(t)|$ can be for this operator to be hyponormal and we will answer this question for large values of $t$. We also correct a typo from a 2019 paper of Fleeman and Liaw concerning the norm of the commutator of the Toeplitz operator with symbol $z^m\bar{z}^n$ when $m>n$.