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 ∂D.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.

Locations

• Indiana University Mathematics Journal - View - PDF
• arXiv (Cornell University) - View