Dual TOEPLITZ Operators On The Orthogonal Complement Of The FOCK Space

This paper deals with the dual Toeplitz operators on the orthogonal complement of the Fock space.Some necessary and sufficient conditions for boundedness and compact-ness of the dual Toeplitz operators are discussed.We also study the structure of the dual Toeplitz algebra and some spectral properties of the dual Toeplitz operators.In Chapter 1,we summarized the significance of the research and the related research ground of the Toeplitz operaors,Hankel operators and dual Toeplitz operaors.We also discuss the basic structure and properties of the Fock space L_α~2(C).In Chapter 2,some basic properties of Toeplitz operators,Hankel operators and dual Toeplitz operaors are discussed.We characterize the boundedness and the compactness of the dual Toeplitz operators.We prove that the dual Toeplitz operator with square-integrable symbol is bounded if and only if its symbol is essentially bounded(Theorem2.4),and that the only compact dual Toeplitz operator is the zero operator(Theorem 2.6).In Chapter 3,we concentrate at a symbol map on the dual Toeplitz algebra.We establish a structure theorem for the dual Toeplitz algebra.We prove that there is a contractive C~*-hormomorphism p from the dual Toeplitz algebra(?)(L~∞(C))to L~∞(C),such that p(S_f)=f,for all f∈L~∞(C)(Theomrem 3.3). Thus there is a short exact sequence (0)→J→(?)(L~∞(C))(?)L~∞(C)→(0), where J is the semicommutator ideal of the dual Toeplitz algebra(Theomrem 3.4). Finally we study spectral properties of dual Toeplitz operator.We prove a spectral inclusion theorem and Brown-Halmos Theorem.The former is analogous to the spectral inclusion theomrem of Hartman and Wintner for Toeplitz operators on the Hardy space.