Problems In Hankel Operator And (?)-Operator

In this thesis,we study two important operators which are the Hankel operator and the(?)-operator in several complex variables.We focus on the compactness of the Hankel operator and the closed range property of the(?)-operator.The compactness of the Hankel operator H? is deeply related to the compactness of the(?)-Neumann operator Nq.Catlin and D'ngelo showed that H? with symbols of functions is compact on A2(?)if N1 is compact on L2(0,1)(?).The converse of this result is known as D'Angelo's question which is not fully understood in general.In-spired by D'Angelo's question,we consider the relationship between the compactness of the the Hankel operator and the compactness of the(?)-Neumann operator.In the thesis,we define the Hankel operator with symbols of forms H?q.The main result of this Hankel operator is an equivalence among(1)the compactness of H?q on K(0,q)2(?),(2)the compactness of canonical solution operator(?)*Nq+k+1 on K(0,q+k+1)2(?),and(3)the compactness of Nq+k+1 on L(0,q+k+1)2(?),for q?1.When the Hankel operator acts on holomorphic functions,the compactness of the Hankel opetator on A2(?)is not equivalent to the compactness of Nq+1 on L(0,q+1)2(?).A sufficient condition and a necessary condition of the compactness of the Hankel operator are also given.Furthermore,we prove the localization theorem of the compactness of the Hankel operator H?q.In the Hilbert space approach,the closed range property for an un-bounded closed operator characterizes the range of this operator.Thus it is important to know whether the range of an unbounded operator is closed.In this thesis,we consider the closed range property of(?)-operator which is a closed and densely defined operator.In complex plane,we show that the(?)-operator has closed range property on the punctured disc D*and the annulus AR,endowed with their Poincare metrics respectively.On the unit disc,the punctured disc and the annu-lus,we creatively derive a sufficient condition and a necessary condition to the closed range property of(?)-operator.In addition,we start from the closed range property on the punctured plane C*and a product do-main to study the closed range property on a special Stein domain and prove that there exist a complete Kahler metric such that(?)-operator has no closed range property.Our result is different from Chakrabarti and Shaw's result which is endowed with the non-complete metric.