Some Problems Of Dual Truncated Toeplitz Operators

The motive force of operator theory is mainly to study various differential equations and integral equations.Its point of view is that a function is regarded as an element in the corresponding function space,and thus a transformation is regarded as an operator or mapping between function spaces that transforms elements into elements,So it can be studied as an abstract space.This paper mainly studies some problems of dual truncated Toeplitz operators defined on the orthogonal complementary space of module space.This paper studies some problems of dual truncated Toeplitz operators with the help of the orthogonal complementary space of model space,which lays a foundation for the subsequent study of the theory of dual truncated Toeplitz operators on harmonic function space In addition,the research method of this paper is to explore the unitary equivalence of harmonic Toeplitz operators,the essential spectrum of harmonic Toeplitz operator product of finite number of continuous symbols and the Halmos fifth problem of dual truncated Toeplitz operator version by comprehensively using analytical,algebraic and geometric methods,and prove that there is no analytic subnormal dual truncated Toeplitz operator,The symbolic characterization of the subnormal dual truncated Toeplitz operator is further given.The following important conclusions are obtained:Theorem 2.1.1 Letf∈L² and u,v be internal functions and at least one of them is non-constant.Let P,then the harmonic Toeplitz operator T_f on H_u,v² and the Dual truncated Toeplitz operator Df on（K_θ²）^⊥ are unitary equivalent.Theorem 3.3.1 If f is a non-constant analytic function,then Df is not subnormal.Theorem 3.3.2 If f∈L^∞,the following three statements are equivalent:1.D_f is a subnormal operator;2.D_f is a normal operator;3.There are constants c and d and a real-valued function φ∈L^∞ that satisfies f=cφ+d a.e.