Multiplication by a finite Blaschke product on weighted Bergman spaces: Commutant and reducing subspaces

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Gallardo-Gutiérrez, Eva A. and Partington, Johathan R. (2022) Multiplication by a finite Blaschke product on weighted Bergman spaces: Commutant and reducing subspaces. Journal of Mathematical Analysis and Applications, 515 (1). ISSN 0022-247X

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Abstract

We provide a characterization of the commutant of analytic Toeplitz operators TB induced by finite Blachke products B acting on weighted Bergman spaces which, as a particular instance, yields the case B(z) = z n on the Bergman space solved recently by by Abkar, Cao and Zhu [2]. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces Hp for 1 < p < ∞. Finally, we apply this approach to study reducing subspaces of TB in the classical Bergman space. As a particular instance, we provide a direct proof of a theorem of Hu, Sun, Xu and Yu [18] which states that every analytic Toeplitz operator TB induced by a finite Blachke product on the Bergman space is reducible and the restriction of TB on a reducing subspace is unitarily equivalent to the Bergman shift.


Item Type:Article
Uncontrolled Keywords:Finite Blaschke products; Commutants; Reducing subspaces; Bergman spaces
Subjects:Sciences > Mathematics
Sciences > Mathematics > Mathematical analysis
ID Code:73300
Deposited On:30 Jun 2022 12:06
Last Modified:01 Jul 2022 06:46

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