The Commutant And Similarity Invariant Of Analytic Toeplitz Operators On Bergman Space

Famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operatorwith n+1-Blaschke factors is unitary to n+1 copies of unilateral shift on Hardy space.Von Neumann-Wold Theorem does not hold on Bergman space. So we have to facemany difficulties in researching the commutant of operators on Bergman space. Thecommutant of an analytic Toeplitz operator on the Hardy space has been studied exten-sively in the literature, we mention here the papers(see[1,4,5,6,7,8,]). In particular, J.E.Thomson gave an explicit description of the commutant of T_B when B is a Blaschkeproduct with two zeros. C.C. Cowen described the commutant of T_b for a finiteBlaschke product in terms of the Riemann surface generated by B. Corresponding tovon Neumann-Wold Theorem on Hardy space, whether each analytic Toeplitz operatorwith finite Blaschke factors is similar to finite copies of Bergman shift? This is a basicand natural problem on Bergman space. K.H.Zhu obtained a complete descriptionof the reducing subspaces of multiplication operators M_b on Bergman space inducedby Blaschke product with two zeros using the method of the geodesic midpoint ofthe two zeros of B(z). In [9],J.Y.Hu, S.H.Sun, X.M.Xu and D.H.Yu proved that theanalytic Toeplitz operator with finite Blaschke product symbol on the Bergman spacehas at least a reducing subspace on which the restriction of the associated Toeplitzoperator is unitary equivalent to Bergman shift. In [10], K.Stroethoff and D.C.Zhengdiscussed the bounded property of product of Hankel and Toeplitz operators on theBergman space and answered Sarason's problem. In this paper, using the basis con-structed by Michael Stessin and Kehe Zhu on Bergman space we prove that each analytic Toeplitz operator M_B(z) is similar to n + 1 copies of Bergman shift if andonly if B(z) is a n + 1-Blaschke product. As some applications of the above theorem,we research the commutant of some analytic Toeplitz operators and generalize someresults on Hardy space to Bergman space. From above theorem, using K₀-group term,we characterize similarity invariant of some analytic Toeplitz operators. The main re-sults in the paper are as follows:(1)Let B(z)∈H^∞(D),then M_B～(?)M_z～M_zⁿ⁺¹ if and only if B(z)is a n+1- Blaschke product．(2)Let F∈H^∞(D)and F=B_n^? be its inner-outer factorization．If for some0≠λ∈C,(?)-λis divisible by B_n,where B_n is a finite Blaschke product．Then(3)Let T∈A'(M_zⁿ)∩A'(M_f),f=z^rg,f∈H^∞(D),g(0)≠0．Then T∈A'(M_z^s),where s=g.c.d．(n,r)．(4)Let F_l∈H^∞(D)(l=1,2),F_l=B_nl(?)l be its inner-outer factorization,whereB_nl=sum from j=1 to n (z-a_jl)/(1-(?)_jl^z) and (?)l is invertible,M_F1∈(SI).Then M_F1～M_F2 if and only if(5)Let f be an analytic function in D and continuous on T,andThen T_f is strongly irreducible if and only if A'(T_f)(?)H^∞(D)．...