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Proceedings of the American Mathematical Society

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Weighted holomorphic spaces with trivial closed range multiplication operators

Authors: Kinga Cichon and Kristian Seip
Journal: Proc. Amer. Math. Soc. 131 (2003), 201-207
MSC (2000): Primary 47B38
Published electronically: May 22, 2002
MathSciNet review: 1929039
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Abstract: We deal with the space $H^{\infty}_{v}$ consisting of those analytic functions $f$ on the unit disc $\mathbb{D}$ such that $\Vert f\Vert _v := \sup_{z \in \mathbb{D} } v(z) \vert f(z) \vert<\infty$, with $v(z)=v(\vert z\vert)$. We determine the critical rate of decay of $v$ such that the pointwise multiplication operator $M_{\varphi}$, $M_{\varphi}(f)(z)=\varphi(z) f(z)$ and $\varphi$ analytic, has closed range in $H^{\infty}_{v}$ only in the trivial case that $\varphi$ is the product of an invertible function in $H^\infty$ and a finite Blaschke product.

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Additional Information

Kinga Cichon
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Matejki 48/49, 60-769 Poznań, Poland

Kristian Seip
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

Keywords: Weighted Banach space of analytic functions, pointwise multiplication operator, closed range
Received by editor(s): April 17, 2001
Received by editor(s) in revised form: August 21, 2001
Published electronically: May 22, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society