  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 the boundary of the disk.
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.
