Commuting Quasihomogeneous Toeplitz Operator and Hankel Operator on Weighted Bergman Space

Jun Yang

doi:10.1155/2013/408168

Abstract

We characterize the commuting Toeplitz operator and Hankel operator with quasihomogeneous symbols. Also, we use it to show the necessary and sufficient conditions for commuting Toeplitz operator and Hankel operator with ordinary functions.

Highlights

Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1
Let eipθf be a bounded function of quasihomogeneous degree p ≥ 0 and g = ∑k∈Z eikθgk,α(r) ∈ L∞(D, dAα)
Let e−ipθf be a bounded function of quasihomogeneous degree −p < 0 L∞(D, dAα)

Summary

Introduction

Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. Let φ ∈ L1(D, dAα) be a radial function; that is, suppose that φ(z) = φ(|z|), z ∈ D. We define the “radialization” of a function f ∈ L1(D, dAα) by the following: rad (f) (z) Let Rα be the space of weighted square integrable radial functions on D.

Results

Conclusion