The Friedrichs Operator and Circular Domains

TL;DR

This paper investigates the Friedrichs operator in complex analysis, establishing conditions for rank one and demonstrating its application to specific domains relevant in control theory.

Contribution

It provides new criteria for the Friedrichs operator to have rank one based on domain coverings and applies this to important domains like the tetrablock and symmetrized polydisc.

Findings

Friedrichs operator has rank one under certain covering conditions.

Rank one property shown for tetrablock, pentablock, and symmetrized polydisc.

Links domain geometry to operator rank and control theory applications.

Abstract

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.