Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces

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.

19 pages, preliminary version

References in corpus (4)

• Self-adjoint, unitary, and normal weighted composition operators in several variables
• Is the Dirichlet Space a Quotient of $D{A}_n$?
• Isometry invariant Finsler metrics on Hilbert Spaces
• Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions