For the open unit disc $\mathbb{ D}$ in the complex plane, it is well known that if $\phi \in C(\overline{ \mathbb{ D}})$ then its Berezin transform $\widetilde{ \phi}$ also belongs to $C(\overline{ \mathbb{ D}})$. We say that $\mathbb{ D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{ C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.
Categories: math.CV math.FA