Compact Operators via the Berezin Transform

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 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.

15 pages. To appear in Indiana University Mathematics Journal. For more information, see http://math.sfsu.edu/axler/CompactBerezin.html