Multiplication and division by inner functions in the space of Bloch functions

Abstract:

A subspace $X$ of the Hardy space $H^1$ is said to have the $f$-property if $h/I \in X$ whenever $h\in X$ and $I$ is an inner function with $h/I \in H^1$. We let $\mathcal B$ denote the space of Bloch functions and $\mathcal B_0$ the little Bloch space. Anderson proved in 1979 that the space $\mathcal B_0\cap H^ \infty$ does not have the $f$-property. However, the question of whether or not $\mathcal B\cap H^ p$ ($1\le p<\infty$) has the $f$-property was open. We prove that for every $p\in [1,\infty )$ the space $\mathcal B\cap H^ p$ does not have the $f$-property.

We also prove that if $B$ is any infinite Blaschke product with positive zeros and $G$ is a Bloch function with $\vert G(z)\vert \to \infty$, as $z\to 1$, then the product $BG$ is not a Bloch function.