Aleman-Richter-Sundberg's Theorem On $P^t(\mu )$-Spaces

TL;DR

This paper proves a new approximation result for the Cauchy transform of certain measures and uses it to give an alternative proof of a theorem on nontangential limits and invariant subspace indices in $P^t(u)$-spaces.

Contribution

It introduces a novel approximation technique for the Cauchy transform of measures annihilating polynomials, leading to an alternative proof of key theorems in $P^t(u)$-spaces.

Findings

Approximate the Cauchy transform with controlled error outside small capacity sets.

Provide an alternative proof of Aleman-Richter-Sundberg's theorem on nontangential limits.

Analyze the index of invariant subspaces using the new approximation method.

Abstract

Let $\nu$ be a finite complex measure with support in $\bar {\mathbb D}$ and let $\mathcal C\nu$ denote the Cauchy transform of $\nu .$ Suppose that $\nu$ annihilates polynomials in complex variable $z$ and $\nu |_{\partial \mathbb D} = hm,$ where $m$ is the normalized Lebesgue measure on $\partial {\mathbb D}$. We show that, for $\epsilon_0 > 0,$ $m$-almost all $e^{i\theta}\in \partial {\mathbb D},$ and $a > 0,$ when $r$ tends to 1, there exists $E_r \subset B(re^{i\theta}, \frac{1-r}{4})$ with analytic capacity $\gamma (E_r) < \epsilon_0 \frac{1-r}{4}$ such that $|\mathcal C\nu (\lambda) - e^{-i\theta}h(e^{i\theta}) | \le a$ area-almost all $\lambda \in B (re^{i\theta}, \frac{1-r}{4} ) \setminus E_r .$ Using this result, we provide an alternative proof of Aleman-Richter-Sundberg's Theorem on nontangential limits in $P^t(\mu )$-Spaces and the index of invariant subspaces.