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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
Published electronically: March 15, 2001
MathSciNet review: 1823922
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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.

References [Enhancements On Off] (What's this?)

  • 1. S. Axler and Z. Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integral Equation Operator Theory, 14(1991), 1-12. MR 92f:47018
  • 2. B.J. Cole and T.W. Gamelin, Tight uniform algebras and algebras of analytic functions, J. Funct. Anal. 46(1982),158-220. MR 83h:46065
  • 3. Z. Cuckovic, Commutant of Toeplitz operators on the Bergman spaces, Pacific. J. Math., 162(1994), 277- 285. MR 94j:47041
  • 4. B. Khani Robati, On the commutant of certain multiplication operators on spaces of analytic functions, Rendiconti del circolo matematico di Palermo, to appear.
  • 5. K. Seddighi and S. M. Vaezpour, Commutant of certain multiplication operator on Hilbert spaces of analytic functions, Studia Math., 133(2)(1999), 121-130. MR 2000i:47062
  • 6. A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J., 20 (1971), 777- 788. MR 44:4558
  • 7. J. E. Thomson, Intersections of commutants of analytic Toeplitz operators, Proc. Amer. Soc., 52(1975), 305-310. MR 53:3765
  • 8. J. E. Thomason, The commutant of certain analytic Toeplitz operators, Proc. Amer. Soc., 54(1976), 165-169. MR 52:8993

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Additional Information

B. Khani Robati
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran

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

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

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