The spectral analysis of the difference quotient operator on model spaces

We conduct a spectral analysis of the difference quotient operator $Q_{\zeta }^u$, associated with a boundary point $\zeta \in \mathrm{\partial }\mathbb{D}$, 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.