Spaces Of Holomorphic Functions And Operators In Several Complex Variables

In this thesis, we investigate some spaces of holomorphic functions in several complex variables and deal with three types of operators. These spaces of holomorphic functions are well known and the operators discussed here have been studied intensively for a long time. The thesis consists of six chapters.In the first chapter, we give a brief introduction of this thesis including notations, dcfinitions, background and main results.In Chapter 2, inspired by the characterizations of Lipschitz spaces and the equivalent norms on Lipschitz type spaces, weighted Hardy-Bloch type space∧_ω,2^p,(B_n) is defined in the unit ball of Cⁿ. We find that Hardy-Bloch type space∧_ω,2^p2(B_n) is a complement of Lipschitz space satisfying integral mean Lipschitz condition and in fact is the mixed norm space H_{∞,p,φ}(B_n) in some sense. Corresponding to the characterization of Lipschitz type spaces in terms of Poisson transform, the membership of the space∧_ω,2¹(B_n) and∧_ω,2²(B_n) is expressed in terms of Berezin transform, whose kernel is the rate of volume change of automorphism and which plays a important role in function theory and in operator theory. As a important class of functions, we also characterize Hardy-Bloch type space∧_ω,2^p(B_n) in terms of the mean integral for 1≥p＜∞.In Chapter 3, firstly, we investigate the properties of functions in weighted Bergman space A^p(Ω, dv_s) for 0＜p≤+∞and -1＜s＜+∞on bounded symmetric domainΩof Cⁿ. Based the Forelli-Rudin type theorem, we obtain some characterizations of functions in A^p(Ω, dv_s) in terms of a class of linear operators D^α,β, which are the generalization of characterizations in terms of various derivatives in Bergman space A^p(B_n, dr_s) on the unit ball. Furthermore, making use of these characterizations, we extend A^p(Ω,dv_s) to the weighted Bergman spaces A_α,β^p(Ω, dv_s) in a very natural way for 1≤p≤+∞and any real number s, that is, -∞＜s＜+∞. This unified treatment covers some classical Bergman spaces and Besov spaces. The boundedness of Bergman projection operators on A_α,β^p(Ω, dv_s) and the dual of A_α,β^p(Ω, dv_s) are given for 1≤p＜∞. In addition, due to the importance of Carleson measure in the function theory, Carleson measure and vanishing Carleson measure for weighted Bergman spaces A_α,β^p(Ω,dv_s) are characterized in terms of Berezin transforms and Bergman metric balls. Since the spaces of holomorphic functions generalized are new and somc Carleson measures are not finite, we get some meaningful results which extend and even provide new insight to those classical weighted Bergman spaces.In Chapter 4, we study the Bloch type spaces B^α(B_n) and consider the Toeplitz operators T_μ,α on B^α(B_n) in the unit ball of Cⁿ for 1≤α＜2, whereμis a positive Borel measure on B_n. We give the necessary and sufficient conditions for T_μ,α to be bounded or compact on B^α(B_n) which improve the results of T_μ,α on the disk where the conditions is only partial for complex measure on the disk. Therefore, positive Borel measuresμon B_n is completely characterized for which T_μ,α is bounded or compact on the Bloch type spaces B^α(B_n)(1≤α＜2).In Chapter 5, we further discuss the Bergman spaces and consider the question for which square integrable holomorphic functions f and g on the polydisk the densely defined Toeplitz type products T_f T_? are bounded on Bergman spaces A²(Dⁿ), where T_? is Toeplitz type operator whose kernel is not Bergman kernel. The Toeplitz type product operator T_fT_? is similar to the Toeplitz product operator T_fT_?, which is widely studied, which is close related to function theory and is a essential operator in operator theory. The emphasis of study about the Toeplitz type product operator is the boundedness and we discover the necessary and sufficient conditions for the boundedness of T_fT(?) are related to the integral transforms of f and g. We prove results analogous to the Toeplitz product T_fT_? on the polydisk and on the unit ball.In the last chapter, we investigate several spaces of holomorphic functions including Hardy spaces, Bergman spaces and Bloch spaces, andconsider the composition type operator between them which is defined by T_ψ,φ(f)=ψf oφ, which is a generalization of a multiplication operator and a composition operator. In terms of the function property ofφandψ, the necessary and sufficient conditions are given for the composition type operator T_ψ,φ to be bounded or compact from the Hardy space H^p(B_n) to theμ-Bloch space B_μ(B_n) and for the composition type operator T_ψ,φ to be bounded or compact from Bergman space A^p(Dⁿ) to Bloch space B(Dⁿ) respectively.In conclusion, from the discussion of the above chapters, we can well understand the properties of holomorphic functions on the special domains of Cⁿ such as unit ball, bounded symmetric domains and polydisk. In particular, we generalize the definitions of the classical spaces of holomorphic functions and reveal the relations between the deferent spaces of holomorphic functions through the studies of the operators. This also leads to the more understanding about operators.