Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

Sumin Kim,Jongrak Lee

doi:10.1186/s13660-021-02602-1

Abstract

In this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator T_{varphi } on the Bergman space A^{2}(mathbb{D}) with non-harmonic symbols under certain assumptions.

Highlights

Let H be a separable complex Hilbert space and L(H) be the set of bounded linear operators on H
For any φ ∈ L∞(D), the Toeplitz operator Tφ on the Bergman space is defined by Tφf = P(φf ) for f ∈ A2(D) and P is the orthogonal projection that maps L2(D) onto A2(D)
In [8, 9], the authors characterized the hyponormality of Toeplitz operators on the Bergman space with harmonic symbols

Summary

Introduction

Let H be a separable complex Hilbert space and L(H) be the set of bounded linear operators on H. For any φ ∈ L∞(D), the Toeplitz operator Tφ on the Bergman space is defined by Tφf = P(φf ) for f ∈ A2(D) and P is the orthogonal projection that maps L2(D) onto A2(D). The hyponormality of Toeplitz operators on the Hardy space has been developed in [2, 3, 10], and [12]. In [8, 9], the authors characterized the hyponormality of Toeplitz operators on the Bergman space with harmonic symbols. We shall list the well-known properties of Toeplitz operators Tφ on the Bergman space. We briefly summarize a number of partial results relating to the hyponormality of Toeplitz operator with non-harmonic symbols, which have been recently developed in [6] and [14].

If s

Hence Tφ is hyponormal if and only if

If t