The Friedrichs Operator of a Planar Domain

Abstract

Properties of the Friedrichs operator of a planar domain, originally intended for applications to planar elasticity, are related to the geometry and fnction theory of the domain. We show that the spectrum of the Friedrichs operator does not, in general, determine the domain, and we provide the additional necessary complete invariants. An analuysis of the Friedrichs operator of generalized quadrature domains is carried out at the abstract level and on several examples. The study of Toeplitz operator on the Bergman space of the domain comes naturally into focus.