Volterra Type Integration Operators On Spaces Of Holomorphic Functions

As important parts in operator theory,Volterra type integration operators on spaces of holomorphic functions have been attracting many attentions due to the fact that they are closely related to the BMO,Hankel forms and semigroups of composition operators.In this thesis,we study the boundedness,(weak)compactness and strict singularity of Volterra type integration operators acting on various spaces of holomorphic functions.This thesis is organized as follows:In Chapter 1,we introduce some backgrounds and current progresses of Volterra type integration operators on spaces of holomorphic functions.Based on the existing results,we clarify the problems to be studied in this thesis.Chapter 2 is devoted to introduce some basic notions and properties in function spaces and operator theory involved in this thesis.Some useful tools,such as Carleson measure and Khinchine’s inequality,are also given.In Chapter 3,using area methods from harmonic analysis,and duality and factorization tricks for tent spaces of sequences,we characterize the compactness of Volterra type integration operators acting from Bergman spaces into Hardy spaces,and give some estimates for the essential norms.Furthermore,in the Hilbert space case,using the related results for Toeplitz operators,we describe the membership in the Schatten(-Herz)class of these Volterra type operators.Corresponding results are also established for Volterra operators from Hardy spaces to Bergman spaces.In Chapter 4,we consider the Volterra type integration operators on holomorphic Hardy-Carleson type tent spaces.We first establish some equivalent norms and the atomic decomposition for Hardy-Carleson type tent spaces.Then,using these results and multipliers of certain sequence tent spaces,we completely characterize the boundedness and compactness of Volterra operators on Hardy-Carleson type tent spaces.We also prove that the compactness and strict singularity coincide for Volterra operators acting on these spaces.In Chapter 5,a class of generalized integration operators on spaces of vectorvalued analytic functions are studied.For a Banach space X,1 ≤p<∞ and α>-1,we estimate the norms of generalized integration operators from the weak X-valued Bergman space wAαp(X)to the strong X-valued Bergman space Aαp(X).In particular,in the case p>2,we prove that the Volterra operator Jb:wAαp(X)→Aαp(X)is bounded if and only if the operator Jb:Aα2→Aα2 belongs to the Schatten class Sp(Aα2),which is different with the case of composition operators.Corresponding results are also obtained on vector-valued Hardy spaces and Fock spaces.In Chapter 6,we study the Volterra type integration operators on growth spaces and Bloch type spaces.Given a Banach space X,we characterize the boundedness and compactness of integration operators between growth spaces and Bloch type spaces on the open unit ball of X.Moreover,using interpolating sequences for bounded holomorphic functions,we prove that the compactness,weak compactness and strict singularity coincide for integration operators acting between these spaces.We finally study the spectral properties of Volterra operators acting on these spaces.