Commutants and hyper-reflexivity of multiplication operators

Authors: MASOUMEH FAGHIH AHMADI, KARIM HEDAYATIAN

Abstract: We characterize the commutants of some multiplication operators on a Banach space of analytic functions defined on a bounded domain in the plane. Under certain conditions on the symbol of a multiplication operator, we show that its commutant is a set of multiplication operators. This partially answers a question of Axler, Cuckovic and Rao. Next, the hyper-reflexivity of these multiplication operators are proved. The paper is concluded by proving the hyper-reflexivity of the multiplication operators with symbols \varphi (z) = z^k, k=1, 2,... .

Keywords: Bergman space, multiplication operators, hyper-reflexive operator, commutant, hyper-invariant

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