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Volume 35 • Number 2A • 2012
• On the Commutant of Certain Operators in the Bergman Space
Ali Abkar

We study the commutant of analytic multiplication operator $M_{z^2}$ on the weighted Bergman space $A^2_\alpha$. According to a result of Zhu [Reducing subspaces for a class of multiplication operators, J. London Math. Soc. (2) 62 (2000), no.2, 553-568], a bounded linear operator $T$ defined on the standard Bergman space $A^2$ commutes with $M_{z^2}$ if and only if there exist two bounded analytic functions $\phi$ and $\psi$ such that $Tf=\phi f_e+\psi f_o/z$ where $f=f_e+f_o$ is the even-odd decomposition of $f$. We shall prove that this statement holds true in the weighted Bergman space as well.

2010 Mathematics Subject Classification: 47B38 (46E20, 30H20).

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