Aleman-Richter-Sundberg's Theorem On $P^t(\hat{I}¼)$-Spaces

Abstract

Let $\hat{I}½$ be a finite complex measure with support in $\bar{\mathbb{D}}$ and let $\mathcal{C}\hat{I}½$ denote the Cauchy transform of $\hat{I}½.$ Suppose that $\hat{I}½$ annihilates polynomials in complex variable $z$ and $\hat{I}½{|}_{\partial \mathbb{D}}=hm,$ where $m$ is the normalized Lebesgue measure on $\partial \mathbb{D}$. We show that, for $\hat{I}{\mu }_0>0,$ $m$-almost all $e^{i\hat{I}¸}\in \partial \mathbb{D},$ and $a>0,$ when $r$ tends to 1, there exists $E_r\subset B(r{e}^{i\hat{I}¸},\frac{1-r}{4})$ with analytic capacity ${\hat{I}}^3(E_r)<\hat{I}{\mu }_0\frac{1-r}{4}$ such that $|\mathcal{C}\hat{I}½(\hat{I}»)-e^{-i\hat{I}¸}h(e^{i\hat{I}¸})|\le a$ area-almost all $\hat{I}»\in B(r{e}^{i\hat{I}¸},\frac{1-r}{4})\setminus E_r.$ Using this result, we provide an alternative proof of Aleman-Richter-Sundberg's Theorem on nontangential limits in $P^t(\hat{I}¼)$-Spaces and the index of invariant subspaces.

7 pages. The results in this papers are special cases of the submitted paper: arXiv:1712.02953 [math.FA]