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

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 $H^{\mathrm{\infty }}(D)$. 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.