 
Summary: Compact Operators via the Berezin Transform
Sheldon Axler Dechao Zheng
25 July 1998
Abstract
In this paper we prove that if S equals a finite sum of finite products
of Toeplitz operators on the Bergman space of the unit disk, then
S is compact if and only if the Berezin transform of S equals 0 on
D. This result is new even when S equals a single Toeplitz operator.
Our main result can be used to prove, via a unified approach, several
previously known results about compact Toeplitz operators, compact
Hankel operators, and appropriate products of these operators.
1 Introduction
Let dA denote Lebesgue area measure on the unit disk D, normalized
so that the measure of D equals 1. The Bergman space L2
a is the Hilbert
space consisting of the analytic functions on D that are also in L2(D, dA).
For z D, the Bergman reproducing kernel is the function Kz L2
a such
that
f(z) = f, Kz
