Density Property and Composition Operators on $H(b)$ Spaces of Finitely Connected Planar Domains

TL;DR

This paper investigates the density of algebraic structures and the boundedness of composition operators on $H(b)$ spaces within finitely connected planar domains, extending known results from the unit disk case.

Contribution

It characterizes the boundedness of composition operators with generalized Blaschke symbols on $H(b)$ spaces over finitely connected domains, and analyzes density properties for extreme and non-extreme points.

Findings

Density of $ ext{A}(D)$ varies with the extremity of $b$ in $H^(D)$

Boundedness of composition operators is characterized for generalized Blaschke symbols

Results extend the unit disk case to finitely connected planar domains

Abstract

In this work, the density in $H(b)$ spaces of finitely connected planar domains and the boundedness of composition operators on these function spaces are studied. Density of the algebra $\mathcal{A}(D)$ is considered for both in the cases where the defining function $b$ is an extreme and non-extreme point of the unit ball of $H^\infty(D)$. In the last part boundedness of composition operators on $H(b)$ spaces is considered and as well as a generalization of the unit disk case is given, the boundedness of composition operators with generalized Blaschke symbols over finitely connected domains is characterized.