The Friedrichs operator and circular domains
Sivaguru Ravisankar, Samriddho Roy · Proceedings of the American Mathematical Society · 2023
The Friedrichs operator of a domain (in C n \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 μ \mu -synthesis in control theory.