Dirichlet空间上乘法算子的约化子空间

Reducing Subspace for a Class of Multiplication Operators on the Dirichlet Space

研究了加权Dirichlet空间乘法算子的约化子空间问题,证明了对于D_α上二重移位M_z^2权系数{β_n}是I-型的,给出了当φ是二阶Blaschke乘积时,乘法算子M_φ在D_α上有且仅有2个非平凡极小约化子空间,不存在极小约化子空间的条件。此外,还给出了在D_α上,一个以2阶Blaschke乘积为符号的乘法算子M_φ与二重移位算子M_z^2酉等价的刻画,丰富了有关解析函数空间上的乘法算子的约化子空间问题的研究成果。

We study the reducing subspace of multiplication operators on the weighted Dirichlet space.We first prove that if Mz2 is a Dirichlet shift with multiplicity 2 on Dαand the weight coefficient{βn}is of type I,whenφis a Blaschke product of 2 order in the unit disk,the multiplication operator Mφonly has 2 non-trivial minimal reducing subspaces and there is no condition for minimal reducing subspace.Besides,we characterize the unitary equivalence on Dαof Mφand Mz2.This paper enriches the research fruits of the reducing subspace problem of the multiplication operator in analytic function space.