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Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface (1806.10757v1)
Published 28 Jun 2018 in math.FA, math.CV, and math.OA
Abstract: This paper is devoted to the study of reducing subspaces for multiplication operator $M_\phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_\phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_\phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $\phi{-1}\circ\phi$. By this means, we determine the reducing subspaces of $M_\phi$ on the Dirichlet space when the order of $\phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang \cite{DPW12}.