Commuting Separately Quasihomogeneous Small Hankel Operators on Pluriharmonic Bergman Space

Qi Wu,Chuntao Qin,Yong Chen,Yukai Zhao

doi:10.1155/2018/5316275

Abstract

We study (semi)commutativity of small Hankel operators with separately quasihomogeneous symbols on the pluriharmonic Bergman space of the unit ball. Some product problems are also concerned.

Highlights

Let Bn be the unit ball in Cn and its boundary Sn
It is well known that L2a(Bn) is a reproducing function space with the reproducing kernel: Kz (w)
Before we prove the last two main results, we should note that, for every small Hankel operator Hh with h ∈ L∞, it is easy to check that Hh is a complex symmetric operator with complex conjugate C which is defined as Cf(z) = f(z) for f ∈ L2(Bn, dV), that is, Hf∗ = CHhC

Summary

Introduction

Let Bn be the unit ball in Cn and its boundary Sn. Let dV denote the normalized Lebesgue volume measure on the unit ball Bn. We study (semi)commutativity of small Hankel operators with separately quasihomogeneous symbols on the pluriharmonic Bergman space of the unit ball. If φ is a bounded separately radial function, We note that it is still open when two Toeplitz operators with separately quasihomogeneous symbols commute.

Results

Conclusion