Bad boundary behavior in star invariant subspaces I

TL;DR

This paper investigates the boundary behavior of functions in certain invariant subspaces of the Hardy space, focusing on growth rates at spectral points where the associated Blaschke product lacks a Carathéodory derivative.

Contribution

It extends previous results by analyzing boundary growth of functions in backward shift invariant subspaces related to Blaschke products without Carathéodory derivatives.

Findings

Identifies conditions for boundary growth rates at spectral points.

Extends Ahern and Clark's results to more general boundary behaviors.

Provides new insights into the boundary behavior of functions in invariant subspaces.

Abstract

We discuss the boundary behavior of functions in backward shift invariant subspaces $(B H^2)^{\perp}$, where $B$ is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of $B$ where $B$ does not admit a derivative in the sense of Carath\'eodory.