1998/07/27 by Sheldon Axler, Dechao Zheng · 1 citation
Mathematics · #math.FA #msc:47B35 #msc:46E20
published as Indiana Univ. Math. J. 47 (1998), 387-400 · 15 pages. To appear in Indiana University Mathematics Journal. For more information, see http://math.sfsu.edu/axler/CompactBerezin.html
arxiv created 1998/07/27 · arxiv updated 2009/11/30
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.