Schur-Nevanlinna parameters, Riesz bases, and compact Hankel operators on the model space

TL;DR

This paper explores the relationship between Schur--Nevanlinna parameters, Riesz bases, and compact Hankel operators in model spaces, providing new constructions and criteria for completeness and basis properties.

Contribution

It introduces methods to construct inner functions with prescribed Schur--Nevanlinna parameters and establishes a compactness criterion for Hankel operators, advancing understanding of basis properties in model spaces.

Findings

Constructed inner functions with specified Schur--Nevanlinna parameters.

Provided a compactness criterion for Hankel operators with specific symbols.

Characterized Riesz bases in relation to boundary behavior of inner functions.

Abstract

We study Riesz bases/Riesz sequences of reproducing kernels in the model space $K_\theta$ in connection with the corresponding Schur--Nevanlinna parameters and functions. In particular, we construct inner functions with given Schur--Nevanlinna parameters at a given sequence $\Lambda$ such that the corresponding systems of projections of reproducing kernels in the model space are complete/non complete. Furthermore, we give a compactness criterion for Hankel operators with symbol $\theta\overline B$, where $\theta$ is an inner function and $B$ is an interpolating Blaschke product and use this criterion to describe Riesz bases $\mathcal K_{\Lambda, \theta}$, with $\lim_{\lambda \in \Lambda, |\lambda| \to 1} \theta(\lambda) =0$.