Hyponormal dual Toeplitz operators on the orthogonal complement of the Harmonic Bergman space

TL;DR

This paper investigates the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space, establishing conditions for hyponormality and properties of their commutators.

Contribution

It provides a characterization of hyponormal dual Toeplitz operators with bounded symbols and analyzes the rank of their commutators in finite rank cases.

Findings

Dual Toeplitz operator with bounded symbol is hyponormal iff it is normal.

Necessary and sufficient condition for hyponormality of specific dual Toeplitz operators.

The rank of the commutator of two such operators is always even if finite.

Abstract

In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator $S_\varphi$ with the symbol $\varphi(z) = az^{n_1}\overline{z}^{m_1} + bz^{n_2} \overline{z}^{m_2}$, $(n_1,n_2,m_1,m_2\in \mathbb {N}$ and $a,b \in \mathbb{C})$ to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.