The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols

TL;DR

This paper investigates the spectral properties of Bergman Toeplitz operators with harmonic polynomial symbols, revealing new phenomena about their spectra, including connectedness and isolated points, and applies these findings to non-hyponormal operators where Weyl's theorem holds.

Contribution

It introduces novel spectral phenomena for Toeplitz operators with harmonic polynomial symbols, including conditions for connected spectra and the existence of isolated spectral points.

Findings

Spectrum of ${\bar{z}+p}$ is connected for degree less than 3.

Existence of polynomials with degree > 2 where the spectrum has isolated points.

Application to non-hyponormal Toeplitz operators satisfying Weyl's theorem.

Abstract

In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On the one hand, we prove that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${\bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.