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

Authors:	Daniel Girela Cristó bal Gonzá lez José Á ngel Pelá ez

Institution:	Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain ; Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain ; Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain

Abstract:	A subspace of the Hardy space is said to have the -property if $h/I \in X$ whenever $h\in X$ and 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 $\mathcal B_0\cap H\sp \infty$ does not have the -property. However, the question of whether or not $\mathcal B\cap H\sp p$ ( $1\le p<\infty$ ) has the -property was open. We prove that for every $p\in 1,\infty )$ the space $\mathcal B\cap H\sp p$ does not have the -property. We also prove that if is any infinite Blaschke product with positive zeros and is a Bloch function with $\vert G(z)\vert \to \infty$ , as $z\to 1$ , then the product is not a Bloch function.

Keywords:	Bloch functions inner functions Blaschke products the $f$-property the $K$-property Toeplitz operators

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