非交换Lebesgue空间上的连续多线性Schur乘子理论及应用

The theory of noncommutative spaces has become a hot research topic in the area of noncommutative analysis. As part of the core contents of noncommutative space theory, Schur multipliers have received a lot of attention due to their important applications in other fields such as the study of differentiability of operator functions, the theory of multiple operator integrals and perturbation theory. Based on the work of the applicant about the Schur multipliers on classical sequence spaces and Lebesgue spaces, this project aims to establish and develop a theory of continuous multilinear Schur multipliers on noncommutative Lebesgue spaces. In particular, we shall extend the Schur multiplier theorem to the noncommutative Lebesgue spaces, generalize the original multiple operator integrals to the noncommutative case and describe the completely boundedness of multiple operator integrals and so on. To these end, we need to investigate new results of factorizations of operators and apply some novel noncommutative analytical techniques. The results to be obtained will become vital tools in the theory of operators and will promote the developments of multiple operator integrals and perturbation theory.

非交换空间理论是非交换分析领域的研究热点。作为非交换空间理论的核心内容之一，Schur乘子理论因其在其他领域(如算子函数的可微性、多重算子积分理论、微扰理论等)的重要应用获得人们大量的关注。本项目拟在申请人已有关于经典序列空间和经典Lebesgue空间上的Schur乘子的工作基础上，建立和发展非交换Lebesgue空间上的连续多线性Schur乘子理论，主要内容包括证明非交换Lebesgue空间上的Schur乘子定理、将原有的多重算子积分理论推广至非交换Lebesgue空间中、刻画多重算子积分的完全有界性等。为此，我们需要建立新的算子分解，同时需要运用一些新奇的非交换分析技巧。该项目所得结果将成为算子理论中的重要工具，并推动多重算子积分理论、微扰理论等进一步发展。

非交换空间理论是非交换分析领域的研究热点。作为非交换空间理论的核心内容之一，Schur乘子理论因其在其他领域(如算子函数的可微性、多重算子积分理论、微扰理论等)的重要应用获得人们大量的关注。本项目围绕非交换Schur乘子展开研究，得到以下两个主要成果：(1)刻画了多重算子积分的完全有界性，该结果推广了Juschenko、Todorov、Turowska的结果；(2) 证明了一个具体算子函数的高阶可微性，作为应用得到了一大类算子函数的算子泰勒余项。相关结果发表于 Canad. Math. Bull.和Rev. Mat. Iberoam.。

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