The restriction operator on Bergman spaces

TL;DR

This paper investigates the restriction operator on Bergman spaces, linking it to Toeplitz operators, analyzing its properties on symmetric subdomains, and providing norm estimates and compactness criteria.

Contribution

It introduces a novel connection between restriction and Toeplitz operators, and offers new norm estimates and spectral invariance results for symmetric subdomains.

Findings

Relation between restriction and Toeplitz operators established

Sharp norm estimates for restriction operator on balls derived

Operator properties like compactness analyzed

Abstract

We study the restriction operator from the Bergman space of a domain in $\mathbb{C}^n$ to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.