Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces

TL;DR

This paper characterizes Hermitean metrics on complex domains derived from Reproducing Kernel Hilbert Spaces and investigates their invariance under automorphisms, revealing that only trivial metrics are invariant under all automorphisms.

Contribution

It provides a formula for Hermitean metrics from RKHSs and characterizes when these metrics are invariant under domain automorphisms, especially relating to the Bergman metric analogue.

Findings

Only trivial metrics are invariant under all automorphisms.

Characterization of RKHSs with automorphism-invariant Bergman metric analogue.

Formulas for metrics derived from RKHS evaluation maps.

Abstract

For a Reproducing Kernel Hilbert Space on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pull-backs of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain. First we approach the problem by considering a stronger property, demanding that the original metric on the (dual of) the RKHS is invariant with respect to all (adjoints of) composition operators, induced by automorphisms. However, we show that only the trivial metric satisfies this property. Then we characterise RKHS's for which the Bergman metric analogue studied in \cite{cd} and \cite{arsw} is automorphism-invariant.