Operator Theory On Bergman Spaces With Exponential Weights And Generalized Fock Spaces

Operator theories on Bergman spaces and Fock spaces have always been one of the research hotspots in the field of functional analysis.These theories are widely used in the fields of harmonic analysis,cybernetics,probability theory,quantum information and complex dynamical systems.In this thesis,we mainly study the boundedness,compactness,Schatten p-class and the invertibility of several classes of classical operators on Bergman spaces with exponential weights and generalized Fock spaces.In Chapter 1,we first introduce the background and development related to this thesis from the classical Bergman space and Fock space.Then we introduce the Bergman space with exponential weight and generalized Fock space and give some basic concepts on these two spaces,respectively.Finally,we state the main results of this thesis.In Chapter 2,we study Toeplitz operators Tμ and weighted composition operators Wψ,φ on the Bergman space with exponential weight Aφp,where μ is a finite positive Borel measure on D,ψ is an analytic function on D and φ is an analytic self-map of D.For 0<p≤∞ and 0<q<∞,we first characterize the(vanishing)q-Carleson measure for Aφp in terms of the averaging function and the generalized Berezin transform.Using these results,we characterize the boundedness and compactness of Toeplitz operators Tμ:Aφp→Aφq for 0<p,q<∞ and weighted composition operators Wψ,ψ:Aφp→Aφq for 0<p≤∞,0<q<∞.Besides,we also characterize Schatten p-class Toeplitz operators and weighted composition operators on Aφ2.In Chapter 3,we study Toeplitz operators Tf induced by f∈IMOs(1<s≤∞),reverse Carleson measures,Toeplitz products and Hankel products on generalized Fock space.Firstly,we characterize the boundedness and compactness of Toeplitz operators Tf:Fφp→Fφq for all 1<p,q≤∞.We characterize reverse Carleson measures for Fφp and then we obtain several equivalent characterizations for invertible Toeplitz operators Tψ induced by positive bounded symbols ψ on Fψ2.Finally,for the special case of generalized Fock spaces,i.e.,the Fock Sobolev spaces F2,m,we investigate the boundedness of Toeplitz products TfTg and Hankel products Hf*Hg on F2,m for two polynomials f and g in z,z∈Cn.