  The Friedrichs operator of a domain (in $\mathbb{C}^n$) is closely related to
its Bergman projection and encodes crucial information (geometric, quadrature,
potential theoretic etc.) about the domain. We show that the Friedrichs
operator of a domain has rank one if the domain can be covered by a circular
domain via a proper holomorphic map of finite multiplicity whose Jacobian is a
homogeneous polynomial. As an application, we show that the Friedrichs operator
is of rank one on the tetrablock, pentablock, and the symmetrized polydisc -
domains of significance in the study of $\mu$-synthesis in control theory.
