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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Daniel Girela, Cristóbal González and José Ángel Peláez
Journal: Proc. Amer. Math. Soc. 134 (2006), 1309-1314
MSC (2000): Primary 30D45, 30D50, 30D55
Published electronically: October 4, 2005
MathSciNet review: 2199173
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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\sp \infty $ does not have the $f$-property. However, the question of whether or not $\mathcal B\cap H\sp p$ ( $1\le p<\infty $) has the $f$-property was open. We prove that for every $p\in [1,\infty )$ the space $\mathcal B\cap H\sp p$ 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.


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Additional Information

Daniel Girela
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
Email: girela@uma.es

Cristóbal González
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
Email: cmge@uma.es

José Ángel Peláez
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
Email: pelaez@anamat.cie.uma.es

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08049-4
PII: S 0002-9939(05)08049-4
Keywords: Bloch functions, inner functions, Blaschke products, the $f$-property, the $K$-property, Toeplitz operators
Received by editor(s): April 22, 2004
Received by editor(s) in revised form: November 17, 2004
Published electronically: October 4, 2005
Additional Notes: This research has been partially supported by a grant from “La Junta de Andalucía” (FQM-210) and by an MCyT grant BFM2001-1736, Spain.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society