The spectral analysis of the difference quotient operator on model spaces

TL;DR

This paper analyzes the spectral properties of the difference quotient operator on model spaces, providing bounds for its norm and insights into the one-component property of inner functions.

Contribution

It offers a detailed spectral analysis of the difference quotient operator on model spaces, including new bounds and characterizations related to inner functions.

Findings

Spectrum description of the operator

Upper and lower norm estimates

New characterization of the one-component property

Abstract

We conduct a spectral analysis of the difference quotient operator $Q^u_\zeta$, associated with a boundary point $\zeta \in \partial \mathbb{D}$, on the model space $K_u$. We describe the operator's spectrum and provide both upper and lower estimates for its norm, and furthermore discussing the sharpness of these bounds. Notably, the upper estimate offers a new characterization of the one-component property for inner function.