The pluriharmonic Hardy space and Toeplitz Operators

TL;DR

This paper introduces the pluriharmonic Hardy space $h^{2}(\mathbb{T}^{2})$ and investigates (semi-)commuting Toeplitz operators with bounded pluriharmonic symbols, revealing novel properties distinct from classical spaces.

Contribution

It defines a new function space and analyzes Toeplitz operators on it, providing new insights into their commutation properties and differences from existing spaces.

Findings

Distinct properties of Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ compared to classical spaces

New characterization of (semi-)commuting Toeplitz operators with pluriharmonic symbols

Methodology offering insights into harmonic Bergman space operators

Abstract

Compared with harmonic Bergman spaces, this paper introduces a new function space which is called the pluriharmonic Hardy space $h^{2}(\mathbb{T}^{2})$. We character (semi-) commuting Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ with bounded pluriharmonic symbols. Interestingly, these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces, Bergman spaces and harmonic Bergman spaces. Our method for Toeplitz operators on $h^{2}(\mathbb{T}^{2})$ gives new insight into the study of commuting Toeplitz operators on harmonic Bergman spaces.