Reducing subspaces for analytic multipliers of the Bergman space

By Ronald G. Douglas, Mihai Putinar, Kai Wang. 2011

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Abstract In Douglas et al. (2011) [4] some incisive results are obtained on the structure of the reducing subspaces for the multiplication operator M ϕ by a finite Blaschke product ϕ on the Bergman space on the unit disk. In particular, the linear dimension of the commutant, A ϕ = { M ϕ , M ϕ ⁎ } ′ , is shown to equal the number of connected components of the Riemann surface, ϕ − 1 ∘ ϕ . Using techniques from Douglas et al. (2011) [4] and a uniformization result that expresses ϕ as a holomorphic covering map in a neighborhood of the boundary of the disk, we prove that A ϕ is commutative, and moreover, that the minimal reducing subspaces are pairwise orthogonal. Finally, an analytic/arithmetic description of the minimal reducing subspaces is also provided, along with the taxonomy of the possible structures of the reducing subspaces in case ϕ has eight zeros. These results have implications in both operator theory and the geometry of finite Blaschke products.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ac877e219386b1a21330400125c85ee9d2b0633d