Compact Operators via the Berezin Transform
Functional Analysis
2007-05-23 v1
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.
Keywords
Cite
@article{arxiv.math/9807147,
title = {Compact Operators via the Berezin Transform},
author = {Sheldon Axler and Dechao Zheng},
journal= {arXiv preprint arXiv:math/9807147},
year = {2007}
}
Comments
15 pages. To appear in Indiana University Mathematics Journal. For more information, see http://math.sfsu.edu/axler/CompactBerezin.html