Summary: Compact Operators via the Berezin Transform
Sheldon Axler Dechao Zheng
25 July 1998
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.
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
f(z) = f, Kz