  Let G be a bounded Jordan domain in the complex plane and consider the
infinite upper Hessenberg matrix M associated with the Bergman orthogonal
polynomials of G. This matrix represents the Bergman shift operator of G. The
main purpose of the paper is to describe and analyze a close relation between M
and the Toeplitz matrix with symbol the normalized conformal map of the
exterior of the unit circle onto the complement of the closure of G. Our
results are based on the strong asymptotics of the Bergman polynomials. As an
application, we describe and analyze an algorithm for recovering the shape of G
from its area moments.
