Toeplitz operators with pluriharmonic symbol

TL;DR

This paper characterizes Toeplitz operators with pluriharmonic symbols on a class of analytic Hilbert spaces on the unit ball, extending classical results from the Hardy and Bergman spaces to higher dimensions.

Contribution

It provides an algebraic characterization of Toeplitz operators with pluriharmonic symbols on $H_m(B)$, generalizing known results from one-dimensional spaces.

Findings

Characterization extends Brown-Halmos theorem to pluriharmonic symbols.

Generalizes Louhichi and Olofsson's results to higher dimensions.

Provides algebraic identity for Toeplitz operators on $H_m(B)$.

Abstract

Let $m \geq 1$ be an integer and let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ given by the reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. We prove that Toeplitz operators with pluriharmonic symbol on $H_m(\mathbb B)$ can be characterized by an algebraic identity which extends the classical Brown-Halmos characterization of Toeplitz operators on the Hardy space of the unit disc as well as corresponding results of Louhichi and Olofsson for Toeplitz operators with harmonic symbol on weighted Bergman spaces of the disc.