The Fock Space Of The Operator And Boundary Representation

The thesis mainly concerns translation operators and boundary representations on the Fock type spaces F_s(0＜s≤1).Fock type spaces are analytic Hilbert spaces over the typical unbounded domain Cⁿ,which generalize the classical Fock space.Operator theory and operator algebras on the Fock type spaces have a deep mathematical and physical background.This thesis considers the Fock type spaces and operator theory and operator algebras on them in the following four aspects.In Chapter One,we mainly consider composition operators on the Fock type spaces,and give a complete characterization of bounded and compact composition operators on a class of Fock type spaces.The results reveal that,compared with those on analytic Hilbert spaces over bounded domains,composition operators on Fock type spaces have much simpler structures.Then a class of natural bounded operators,namely the translation operators on the Fock type spaces F_s(0＜s≤1), are obtained.In Chapter Two,under the framework of analytic Hilbert modules,we consider the classification of translation invariant subspaces of the Fock type spaces up to unitary equivalence.It is showed that under the module action induced by the tuple (T₁,…,T_n) of translation operators,the submodules and quotient modules of F₁ are all rigid.In Chapter Three,using Arveson's boundary representation theory,we study operator algebras generated by translation operators on the Fock type spaces F_s(0＜s≤1).The main problem in concern is that whether the identity representation of the C^*-algebra C^*(T1,…,T_n) is a boundary representation for the Banach algebra B(T₁,…,T_n).It turns out that on the space F₁,the answer is No; while on the space F_s(0＜s＜1),the answer is Yes.Boundary representations on submodules and quotient modules of the space F₁ are also considered.Through a unitary equivalence relation,it is converted into a problem about boundary representations on quotient modules and submodules of a weighted Bergman space.In the one-dimensional case,we give a necessary and sufficient condition for boundary representations on the quotient modules of the Bergman space.In the higher dimensional case,we give a sufficient condition,and also point out the connection between boundary representations and essential normality of modules.In Chapter Four,we use the Fock type space F₁ as a model to study the invariant subspace problem.By exploring maximal invariant subspaces of a class of operators,we obtain a general conclusion.Applications of the general conclusion to the Bergman shift operator and translation operators on the space F₁ give out a series of interesting and deep results,and also enable us to point out the obstacle of the invariant subspace problem.