# One-component inner functions

## TL;DR

This paper identifies specific classes of inner functions in $H^$ with connected sublevel sets, highlighting their significance in operator theory.

## Contribution

It explicitly characterizes several classes of one-component inner functions with connected level sets in the unit disk.

## Key findings

- Identified classes of inner functions with connected level sets.
- Connected level sets occur for specific inner functions.
- Highlights importance of one-component inner functions in operator theory.

## Abstract

We explicitely unveil several classes of inner functions $u$ in $H^\infty$ with the property that there is $\eta\in ]0,1[$ such that the level set $\Omega_u(\eta):=\{z\in\mathbb D: |u(z)|<\eta\}$ is connected. These so-called one-component inner functions play an important role in operator theory.