The Hardy-Type Toeplitz Operators On The Dirichlet Space

Operator theory is one of the main branches of functional analysis.In this paper,we study basic propeties,compactness and algebra properties of Hardy-Type Toeplitz operators which are induced by Szego projection and bounded harmonic functions on the Dirichlet space.First of all,we characterize the compactness of the Hardy-Type Toeplitz operators,extend the consequence of compact operators on the Hardy space,and obtain this kind of operators are compact if and only if its symbol functions are zero.Also,we give the conditions of symbol functions when the product of two Hardy-Type Toeplitz operators is still a Toeplitz operator,and compute the essential spectrum by building a peak function.Next,we calculate matrices of these Hardy-Type Toeplitz operators with respected to the orthonormal basis.We use the symmetry,normality and commutativity of the matrices of this kind of operators to study the basic properties of Hardy-Type operators,including self adjointness,normality,commutativity and invertibility.