Hankel算子的代数运算性质

Hankel算子是函数空间上的一类重要算子，与很多数学分支紧密相连，例如Riemann猜测与Hankel算子的奇异值密切相关。虽然关于函数空间上的Hankel算子的研究已超过六十年，但至今仍有一些问题很难攻克，而且还不断产生新的问题及应用。本项目将综合运用函数论、算子理论及算子代数的方法与技巧来研究：Hardy空间上两个Hankel算子乘积是迹类或Schatten-p类的刻画；两个Hankel算子与一个Toeplitz算子的混合交换性以及一个Hankel算子与两个Toeplitz算子的交换性刻画。这些问题中有些是经典理论中留下的难点，有些产生着新的应用，对于充实函数空间上的算子理论和促进其发展有重要的科学意义。

Hankel operators are an important class of operators on function spaces, which are closely related to many fields of Mathematics, for example, it was proved that the Riemann Hypothesis is closed related to the eigenvalues of Hankel operators. Although the study of Hankel operators on function spaces has a long history over sixty years, there are still some problems on Hankel operators which are difficult to be worked out completely until now, moreover, more and more new questions and applications about Hankel operators appear. This project is aimed to study some questions about the Hankel operators by combining the techniques of function theory, operator theory and operator algebras, which is expected to solve the problem of when the product of two Hankel operators on Hardy space is in the trace classes or in the Schatten-p classes; the problem of the mixed commutativity of two Hankel operators and a Toeplitz operator on Hardy space or the mixed commutativity of a Hankel operator and two Toeplitz operators. Some of the above problems are difficult problems remaining in the classical operator theory until now, and some of them give rise to new applications, thus this project is full of important scientific meanings both on enriching and stimulating the development of the operator theory in function spaces.