Toeplitz Operators And Hankel Operators On Bergman And Fock Spaces

In this dissertation,we study the boundedness of Toeplitz operators with general symbols on Bergman spaces,bounded and compact Hankel products and Toeplitz products on Fock spaces,and the boundedness and compactness of products of Hankel operator and Toeplitz operator on Fock spaces.We obtain a sufficient condition for Toeplitz operator with locally integrable symbols to be bounded on the Bergman spaces_aL^p with 1(27)p(27)?in terms of generalized Carleson squares.It is shown that this condition is equivalent to Taskinen and Virtanen's condition,and they are not necessary.The corresponding result for compactness is also obtained.Moreover,we get a similar result for Toeplitz operators with integrable symbols on Bergman space over upper half plane.On Fock spaceF_?²,forf,g?F_?² we study the boundedness and compactness of Hankel productH_f⁽⁹⁾H_g.This is a companion to Sarason's Toeplitz problem.We will show that the boundedness of functionD(f,g)(z)(28)H_f k_z ²H_g k_z ²is a sufficient but not necessary condition.Actually,there are non-constant functions f and g inF_?²such thatH_f⁽⁹⁾H_g(28)0,while the function D(f,g)is unbounded.However,on Bergman space this condition is necessary for Hankel product to be bounded,and the sufficiency is open.This result is in sharp contrast to the Cho-Park-Zhu solution for Toeplitz products problem in Fock space setting(see[1]).Furthermore,for some special symbol classes we will characterize the boundedness and compactness of Hankel products completely.Another companion to Sarason's Toeplitz product problem is the boundedness and compactness of the products of Hankel operator and Toeplitz operator(Haplitz products).Forf,g?F_?²,we determine exactly when the productH _fT_g of Hankel operatorH_f and Toeplitz operator _gT is bounded or compact on Fock space.We will give an elementary explicit condition for the symbols f and g.On Bergman space,this problem also remains open.Moreover,we characterize bounded Toeplitz product T_f T_g on Fock spaceF_?² for two polynomials f and g in z and z.