On Similarity And Reducing Subspaces Of A Class Of Integral Operator On Dirichlet Spaces

In the field of operator theroy and operator algebra,there is an intresting problem:whether every bounded operator on a complex Banach space can sends some non-trivial closed subspace to itself.It is the well-known "Invariant Subspace Problem" which has not been completely solved.Although some special operator classes in Banach spaces has been partailly solved,it is still an open problem for separable Hilbert spaces.In a sense,shifts are the "Building Block" of operator theory.Many important op-erators are "made up" of shifts.The quasi-nilpotency and compactness of the Volterra operator made it rather widely used.Based on this,many mathmaticians devoted them-selves to the study of this two classes of operators.In the past seventy years,many remarkable achievements have been made on the similarity and reducing subspace of n-shift operators on Bergman Spaces.Also,the similarity and reducing subspace of the Volterra operator has gain lots of attention.However,the study combining these two is really rare.From many aspects,the Dirichlet space is a borderline case which makes it an ex-citing and challenging case in function spaces.Since many important questions are still unsolved,the Dirichlet space remains to be an active area of research,written in A Primer on the Dirichlet Space by Omar El-Fallah et al.in 2014.This dissertation characterizes the similarity and reducing subspaces of n-shift plus certain weighted Volterra operator on the Dirichlet space,aiming at enriching the related results about the reducing subspaces of Volterra operator in analytic function spaces.In the dissertation,let p(z)be a polynomial,it defines the operator T1 to be the operator Mp(z)plus certain weighted Volterra operator.Firstly,it characterizes the similarity of T1.By using some techniques in operator theory it verifis that the action of T1 on the Dirichlet space is similar to the action of multiplication operator Mp on the S(D)space.Furthermore,it proofs that when p(z)equales to zn,the corresponding operator T2 has exactly 2n reducing subspaces.The dissertation is organized as follows:Chapter 1 introduces backgroud about the research.It introduces the origin and present stage of the invariant subspace problem,and summarizes terminologies and results related.Chapter 2 focuses on characterizing the similarity of T1.Firstly,by flexibly using the knowledge of series and Cauchy-Schwarz inequality,it explores the rang of T1.Secondly,according to some techniches in operator theory it constructs the spaceS(D)making it just the rang space of Vp and gives a definition of the norm and inner product of S(D).Thirdly,it characterizes some natures of Vp like boundness,invertibility,similarity and so on,and proofs T1 is similar to Mp.Finally,the essential part of the conclusion can be rephrased into a diagram which is commutative.Chapter 3 studies the reducing subspace of T2.It uses the general-to-specific idea and some tools like commutative algebra,projection operator and block matrix.Firstly,it discusses the representation under an orthonormal basis of a general bounded operator.Secondly,it discusses the representation under the above basis of a projection operator.Finally,through complicated computing of infinite matrix it gives the reducing subspace of T2 is c1 H1(?)c2H2(?)…(?)cnHn,(ci=0,1).Chapter 4 gives a conclusion and outlook of the research.