Reducing Subspace For A Class Of Multiplication Operators On The ?-drichlet Space

In this thesis,we discuss reducing subspace of multiplication operators M? on the?-Dirichlet space D? defined by a finite Blaschke product ?,and prove that the multiplication operators on the Dirichlet space is unitarily equivalent to Dirichlet shift.This thesis is divided into three chapters.In Chapter 1,we discuss some related research background,and give some basic concept and notations.At last,we show the significance of the research work.In Chapter 2,we describe reducing subspace of multiplication operators M? on the a-Dirichlet space D? defined by a Blaschke product ? with two zero in the unit disk D.In the case ?>-1,we prove that multiplication operators M? on the ?-Dirichlet space D? defined by a Blaschke product ? with three zero in the unit disk D is irreducible.In Chapter 3,we describe a multiplication operators M? on the ?-Dirichlet space D?defined by a Blaschke product ? with n+1 zero in the unit disk D is unitarily equivalent to Dirichlet shift of multiplicity n+1.The main results of this article generalize many known results.