非交换算子空间理论是当前泛函分析领域一个活跃的研究方向，将von Neumann代数与经典Hardy空间结合建立非交换Hp空间理论，受到国内外学者的广泛关注。本项目以算子论，算子代数与复分析为理论基础，利用酉不变范数代替p-范数，研究半有限von Neumann代数上的非交换广义Lp空间与非交换广义Hp空间理论。具体而言，借助非交换Banach-Stone定理刻画非交换广义Lp空间的等距同构; 基于Haagerup约化理论，利用算子代数中的范数拓扑结构，研究非交换广义Hp空间中外函数的特征与内外分解的性质，考虑非交换Beurling不变子空间问题；应用Toeplitz算子在次对角代数中的部分等距，研究其在非交换广义Hp空间中的内外分解定理与交换子问题。本项目期望在梳理和发展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 noncommutative operator space is an active research field in functional analysis. Recently, noncommutative Hardy spaces based on von Neumann algebra and classical Hardy spaces have attracted more attention. In this project, from the theory of operator algebra, operator theory and complex analysis, we will deeply study the noncommutative generalized Lp spaces and noncommutative Hp spaces in semifinite von Neumann algebra by replacing p-norms with unitarily invariant norms. Specifically, based on the noncommutative Banach–Stone theorem, we are going to characterize the isometric isomorphism of noncommutative Lp structure. Also, due to Haagerup’s reduction theorem, the topological structures in semifinite von Neumann algebra will help us to investigate the outer functions and inner-outer factorization theorem, then we can extend the Beurling's invariant subspace theorem to the new setting；Meanwhile, by the isometry of Toeplitz operator in Arveson’s noncommutative subdiagonal algebra, we can deduce the inner-outer factorization theorem and the commutant of Toeplitz operator in noncommutative generalized Hardy spaces. In the process of comparing the traditional method with our new idea 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.