article

Title: Two theorems on star-invariant subspaces of BMOA

Authors: Konstantin M. Dyakonov

Issue: Volume 56 (2007), Issue 2, 643-658

Abstract:

$For an inner function $\theta$, let $K^2_{\theta}:=H^2\ominus\theta H^2$ and $K_{* heta}:=K^2_{\theta}\cap\mathrm{BMO}$. Two theorems are proved. The first of these provides a criterion for a coanalytic Toeplitz operator to map $K_{*\theta}$ into a given space $X$, under certain assumptions on the latter. In particular, many natural smoothness spaces are eligible as $X$. As a consequence, for such spaces one has $K_{*\theta}\subset X$ whenever $\theta\in X$. The second theorem concerns the relationship between $K_{*\theta}$ and its counterpart $\widetilde{K_{*\theta}}$, defined as the image of $K_{*\theta}$ under the natural involution $f\mapsto\bar{z}\bar{f}\theta$. Specifically, it is proved that the inner factors associated with the two classes are the same if and only if $\theta$ is a Blaschke product whose zeros satisfy the so-called uniform Frostman condition.$