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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 $phi^{-1}circphi$. 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}.
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_phi$ are orthogonal. When the ord
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ 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 recen
By analytic perturbations, we refer to shifts that are finite rank perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and $F$ is a finite rank operator on the Hardy space over the open unit disc. Here shift refers to the multipl
We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = sum a_n n^{-s-bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert
We obtain sufficient conditions for a densely defined operator on the Fock space to be bounded or compact. Under the boundedness condition we then characterize the compactness of the operator in terms of its Berezin transform.