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On the commutant of operators of multiplication by univalent functions


Authors: B. Khani Robati and S. M. Vaezpour
Journal: Proc. Amer. Math. Soc. 129 (2001), 2379-2383
MSC (2000): Primary 47B35; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-01-05959-7
Published electronically: March 15, 2001
MathSciNet review: 1823922
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Abstract:

Let $\mathcal{B}$ be a certain Banach space consisting of continuous functions defined on the open unit disk. Let ${\phi}\in \mathcal{B}$ be a univalent function defined on $\overline{\mathbf{D}}$, and assume that $M_{\phi}$ denotes the operator of multiplication by ${\phi}$. We characterize the structure of the operator $T$ such that $ M_{\phi} T=T M_{\phi}$. We show that $T=M_{\varphi}$ for some function ${\varphi}$ in $\mathcal{B}$. We also characterize the commutant of $M_{{\phi}^2}$ under certain conditions.


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Additional Information

B. Khani Robati
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran
Email: Khani@math.susc.ac.ir

S. M. Vaezpour
Affiliation: Department of Mathematics, Yazd University, Yazd, Iran

DOI: https://doi.org/10.1090/S0002-9939-01-05959-7
Keywords: Commutant, multiplication operators, Banach space of analytic functions, univalent function, bounded point evaluation
Received by editor(s): December 16, 1999
Published electronically: March 15, 2001
Additional Notes: Research of the first author was partially supported by a national grant (no. 522)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society