Berezin symbols and operator theory

Abstract

Let 'H' be a standard analytic functional Hilbert space over a bounded domain [Omega] ? C. We examine the Berezin symbols 'A~' of bounded operators A?BH and characterize the compact operators KH by Berezin symbol behavior. We show that A?KH iff the Berezin symbol of every unitary conjugate of 'A' is in 'C'0([Omega]) (Nordgren and Rosenthal, 1994). Special attention is also given to examples and the theory of Berezin symbols on the Bergman and Hardy space. We show a characterization (Axler and Zheng, 1998) of compact Toeplitz operators on the Bergman space that generalizes to Hankel operators. The condition 'A' is compact iff A*A&d15;z [right arrow]0 as @'z'@ [right arrow] 1- holds for all Toeplitz, Hankel, and composition operators on both the Bergman and Hardy spaces.