COMPACT-OPERATORS VIA THE BEREZIN TRANSFORM

In this paper we prove that if S equals a finite sum of finite product s 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 parti al derivative D. This result is new even when S equals a single Toepli tz operator. Our main result can be used to prove, via a unified appro ach, several previously known results about compact Toeplitz operators , compact Hankel operators, and appropriate products of these operator s.