Operator Theory On The Fock Space

The Fock space originated from Bargmann’s research on many-body quantum physics,especially on Bose-Fock space.Bargmann’s initial goal was to represent the Bose-Fock space using a function space,then one can represent some operators on the Bose-Fock space to Toeplitz operators on the Fock space.From this idea,a series of studies on operators using function space and function theory have been initiated.This paper includes the following chapters:In the introduction,we will introduce the specific source of the Fock space in detail.We will start with the basic assumptions of quantum mechanics and gradually introduce Bose-Fock space.Based on the Bose-Fock space,we will introduce the Fock space in the last,which described the entire background of the Fock space.In Chapter 2,we will introduce the Fock space onand obtain an isomorphism between the Fock space onand the Bose-Fock space.Based on this isomorphism,we will obtain representations of some operators on the Bose-Fock space and answer a question in[5].Based on the physical meaning of the Bose-Fock space,we will apply this theory to the calculation of the Gibbs state.In Chapter 3,in sprite by Berger-Coburn theorems and their conjecture in[6],we use the Fourier transform to decompose a Toeplitz operator as an infinite sum of Toeplitz operators with symbols which have compact support in the frequency domain.As a consequence,we obtain a sufficient condition for a Toeplitz operator to be bounded in terms of the Carleson measure conditions defined by the heat transform of its symbol,this improves the existing conclusions.Moreover this decomposition of a Toeplitz operator leads us to get easily understanding about properties of the operator.As an example,we will discuss compactness of a Toeplitz operator.In Chapter 4,we study the structure of Toeplitz algebras.Let BUC(C~n)denote the set of bounded uniformly continuous functions on C~n.On the Fock space,we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BU C(C~n)as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols.This answers a question recently posed by Fulsche[12].In Chapter 5,we will study the characterization of Toeplitz algebras.We will introduce the band dominated operators on the Fock space and prove that the operator algebra generated by the band dominated operators is equivalent to the Toeplitz algebra with bounded symbols on the Fock space.Moreover,we will introduce Bergman-type spaces which is a generalization of the Fock space.We will prove that many conclusions can be extended to Bergman-type spaces.