Boundary values of holomorphic functions and some spectral problems for unitary representations

TL;DR

This paper explores boundary value problems for holomorphic functions in Cartan domains, linking them to spectral analysis of unitary representations and Berezin kernels on symmetric spaces.

Contribution

It introduces conditions for restriction operators on holomorphic functions and applies trace theorems to analyze spectra of unitary representations, including singular cases.

Findings

Established boundary value conditions for holomorphic functions in Cartan domains.

Linked restriction operators to spectral increments of unitary representations.

Analyzed spectral problems involving Berezin kernels on Riemann symmetric spaces.

Abstract

We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator exists. Using such 'trace theorems' it is possible to construct discrete increments to spectra of some unitary representation and to catch singular unitary representations in the spectra. We also discuss spectral problems related to Berezin type kernels on riemann symmetric spaces.