Reducing Subspaces For Toeplitz Operators

In the last four decades,many remarkable results of reducing subspaces for multiplication operators on analytic function spaces have been achieved.First,in 1970s,the commutants and reducing subspaces for multiplication operators on the Hardy space of the unit disk have been studied by Nordgren,Thomson,Cowen,et.al.Stimulated by the research at this stage,a natural question is what the results about the reducing subspaces for the operators on other analytic function spaces look like.Except for the classical Hardy space,Bergman space is also a closely watched analytic function space.Recently,some remarkable results about the reducing subspaces for the multiplication operators on the Bergman space over the unit disk have mushroomed,which show the fascinating association between analysis,geometry,algebra,group theory and other branches of mathematics.In this thesis,we mainly study on the structure of the reducing subspaces and relevant commutant algebras for a class of non-analytic Toeplitz operators on the Bergman space over the polydisk.In Chapter 1,the research progress of the reducing subspaces for multiplication operators on analytic function spaces has been introduced.In Chapter 2,the structures of reducing subspaces and commutant algebras for T?z1k+?z2-l(k,l ?Z+,?,??C,??? 0)on the Bergman space over the bidisk have been characterized.First,the direct sum decomposition of the whole space is given and a diagonal operator is con-structed.Furthermore,an equivalence relation is defined due to the weight sequence;Second,the concrete forms of projections and unitary operators are obtained by comparing the coeffi-cients;Finally,all the minimal reducing subspaces are characterized and then the structure of the commutant algebra is given.In Chapter 3,first,if ?(z)= zk?z-l(k,l? Z+),the structure of reducing subspaces for T? on the Bergman space over the bidisk is characterized.If k ? l,we find all the minimal reducing subspaces for T? by constructing the diagonal operator.In addition,the results indicate that there are no other reducing subspaces for T? except for the common reducing subspaces for Tzk and Tzl.Ifk=l,then T? is a normal operator.By the properties of normal operators,we obtained the results which are very different with the case k?l.Moreover,the result for the case k = l is true on the Bergman space over the polydisk.Meanwhile,if ?(z)= Zk +z-l(k,l? Z+),the structure of reducing subspaces for T?,on the Bergman space over the unit disk is also characterized.Furthermore,the results above include the known cases ?=z1Mz2N,? = z1M + z2-N and ? =ZM,M,N ? Z+.Finally,by the structure of reducing subspaces,the structures of V*(?)and V*(?)are both obtained.