Toeplltz operators and the Berezin transform on H2 Original Research Article

For T a bounded linear operator on the Hardy space H2, its Berezin transform is the function image on the unit disc defined by image where image is the normalized reproducing (Szegö) kernel. When T = T/Gf is a Toeplitz operator, it turns out that the nontangential boundary values of image coincide with /Gf almost everywhere. This allows us to construct extensions and/or generalizations of the classical symbol calculus for Toeplitz operators to larger operator algebras. Briefly discussed is also the relation between our results and the recent results of Berger, Coburn and Zhu for Toeplitz and Hankel operators on the Bergman space.