Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surface

TL;DR

This paper investigates the reducing subspaces of multiplication operators on the Dirichlet space with finite Blaschke product symbols, using local inverses and Riemann surfaces, and determines these subspaces for specific cases of order 5, 6, and 7.

Contribution

It introduces a novel approach using local inverses and Riemann surfaces to analyze reducing subspaces on the Dirichlet space, extending previous methods and answering open questions.

Findings

Determined reducing subspaces for orders 5, 6, and 7.

Established a connection between Dirichlet and Bergman space reducing subspaces.

Developed a new method involving Riemann surface analysis for these operators.

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}.