Finite rank perturbations of Toeplitz products on the Bergman space

Abstract

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if Fj,Gj (1≤j≤N) are polynomials of z and z¯ then ∑j=1NTFjTGj−TH is a finite rank operator for some L1-function H if and only if ∑j=1NFj⋄Gj belongs to L1 and H=∑j=1NFj⋄Gj. In the case Fj's are holomorphic and Gj's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.