Products Of Toepltz Operators And Hankel Operators On The Fock Spaces

Operator theory in function spaces is an important component of the operator theory,and its core problem is how to describe the properties of operators themselves by the properties of their symbol functions.In this dissertation,we study the bounded and compact Hankel products on the Fock space and the Fock-Sobolev space,the boundedness and the compactness for the difference Cφ-Cφ of the composition operators Cφ and Cφ on some analytic function spaces,and the boundedness for mixed products of Toeplitz and Hankel operators on the Fock-Sobolev spaces.The Hankel product problem is a companion to Sarason’s Toeplitz product problem.It is still an open problem on Hardy spaces and Bergman spaces as of now.Stroethoff and Zheng proposed a conjecture about the Hankel product problem on the Bergman space in 1999,which is meaningful to the Hardy and Bergman and Fock spaces.Ma-Yan-Zheng-Zhu proved the conjecture prosed by Stroethoff and Zheng was false on the Fock space in 2019 and the condition for the conjecture was sufficient but not necessary on the Fock spaces.The paper proves that the conjecture is true under some condition on the Fock space F2(C).The dissertation also completely characterizes the boundedness and compactness of the Hankel product H(?)*H(?) on the Fock-Sobolev space F2,m(C)for some symbol classes.Moreover,the specific information of the symbol functions f and g will also be given to satisfy that the Hankel product H(?)*H(?) on the Fock space and the Fock-Sobolev space is bounded or compact.Another companion to Sarason’s Toeplitz product problem is the problem for the mixed products of Toeplitz and Hankel operators(Haplitz products),which is also still open on Hardy and Bergman spaces.For f,g ∈ Fp,m(Cn)where 1 ≤p<∞,we determine exactly when the Haplitz product H(?)T(?) is bounded or compact on the Fock-Sobolev spaces and also give an elementary explicit condition for the symbols f and g.Furthermore,this paper also characterizes bounded and compact Haplitz product H(?)T(?) on the Fock-Sobolev spaces Ft p,m(Cn)where t>0.In addition,we also characterize completely the boundedness and compactness for the difference Cφ-Cφ of the composition operators Cφ and Cφ from Bloch space B into Besov space Bv∞.Moreover,we also give a complete characterization of the compactness of the difference Cφ-Cφ on BMOA space.