Paths of inner-related functions

TL;DR

This paper characterizes the structure of certain subsets of bounded analytic functions related to Blaschke products and explores the connectivity of inner functions within these sets, revealing new insights into their topological and algebraic properties.

Contribution

It provides a detailed characterization of connected components in subsets of $H^$ formed by products of Carleson-Newman Blaschke products and invertible functions, and shows how all inner functions relate to these components.

Findings

No inner function in the little Bloch space, except finite Blaschke products, lies in the closure of these components.

Every inner function can be connected to an element of $ ext{cni}$ via products involving inner and invertible functions.

The results extend to Douglas algebras, broadening their applicability.

Abstract

We characterize the connected components of the subset $\cni$ of $H^\infty$ formed by the products $bh$, where $b$ is Carleson-Newman Blaschke product and $h\in H^\infty$ is an invertible function. We use this result to show that, except for finite Blaschke products, no inner function in the little Bloch space is in the closure of one of these components. Our main result says that every inner function can be connected with an element of $\cni$ within the set of products $uh$, where $u$ is inner and $h$ is invertible. We also study some of these issues in the context of Douglas algebras.