经典Hardy空间（交换的Hp空间）理论是复分析的重要组成部分之一。近年来，von Neumann代数上的非交换Hp空间理论受到了国内外学者的广泛关注。本项目以算子论，算子代数与经典复分析为理论基础，以von Neumann代数上的次对角代数为非交换解析模型，利用酉不变范数代替常规的p范数，系统研究非交换广义Hardy空间理论。具体而言，借助广义Hölder不等式，研究非交换Lp空间的对偶，给出非交换广义Hardy空间中解析算子的刻画；利用von Neumann代数中范数拓扑结构理论，探讨非交换广义Hardy空间中Beurling不变子空间问题；利用Fuglede-Kadison 行列式分析解析算子空间中“外函数”的特征，进而解决内外型分解问题。在梳理和发展Hardy空间中传统研究方法的同时，探索非交换Hardy空间研究的新方向和新问题，促进算子论和算子代数的进一步发展。

The theory of classical Hardy spaces is an important part in complex analysis. Recently, noncommutative Hardy spaces based on von Neumann algebra have attracted more attention. In this project, based on the theory of operator algebra, operator theory and complex analysis, starting with the maximal subdiagonal algebra in von Neumann algebra, we will deeply study the noncommutative generalized Hardy spaces by replacing the usual p-norms with unitarily invariant norms. Specifically, by the generalized Hölder's inequality, the dual spaces of noncommutative Lp spaces will be investigated, then we will provide the characterizations of generalized Hardy spaces；The topological structures with respect to unitarily invariant norms in the von Neumann algebra will help us to extend the Beurling's invariant subspace theorem；Using the Fuglede-Kadison determinant, we will study the “outer functions” in the general setting and solve the inner-outer factorization problem. In the process of comparing the traditional methods with our new ideas used in this project, new directions and new problems in the study of Hardy spaces will be explored, which will further the development of operator algebra and operator theory.