Compact Operators via the Berezin Transform

Details Compact Operators via the Berezin Transform Compact Operators via the Berezin Transform

1998-07-27

arxiv.org/abs/math/9807147v1

Indiana Univ. Math. J. 47 (1998), 387-400

Mathematics

Functional Analysis

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

Scientific paper

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.

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