函数空间上拟共形或拟正则符号的复合算子

Composition operator is an important branch of operator theory. Fruitful and profound results have been obtained in the study of composition operators with analytic symbols on function spaces. Recently, much enthusiasm have been shown to the study of composition operators with non-analytic symbols. However, when the symbols are general non-analytic functions, due to the complexity and lack of powerful tools, no profound enough and systematic results have been obtained at present. As generalization of conformal or analytic functions, quasiconformal or quasiregular mapping is the international mainstream research fileld. Methods from real or harmonic analysis other than analytic function theory are mostly used in its research. In this project, we aim to combine methods from operator theory, quasiconformal mapping, classical harmonic analysis to study composition operators with symbols being quasiconformal or quasiregular mappings, acting on classical analytic reproducing kernel Hilbert function spaces over the unit disk or the unit ball such as Hardy space, weighted Bergman space etc. We aim to find out the internal relationship between the properties of composition operators and the analytic or geometric properties of the symbol functions, and investigate the relationship between operator theory and quasiconform mapping.

复合算子是算子理论的重要分支。目前函数空间上解析符号的复合算子的研究，取得了丰富而深入的结果。对非解析符号的复合算子，近年来国内外有很大的研究热情。但当符号函数过于一般时，研究较为复杂而缺少强有力的研究手段，目前还不能取得足够深入而系统的成果。拟共形映射或拟正则映射，作为共形映射或解析函数的推广，是国际上主流的研究方向，其研究更多的采用实分析、调和分析而非复分析的方法。本项目拟采用算子理论、拟共形映射、经典调和分析相结合的方法，致力于研究单位圆盘或单位球上如Hardy空间、加权Bergman空间等经典解析再生核函数空间上的拟共形或拟正则映射为符号的复合算子，力求建立其算子性质与符号函数的分析、几何性质的内在联系，并挖掘算子理论与拟共形映射之间的关联。

解析符号的复合算子的研究取得了丰富的成果，非解析符号的复合算子还缺乏较为系统的研究，选取拟共形映射作为符号函数是一个合适的切入点，郭坤宇教授、方向教授和王子鹏的文章在这方面取得了重要的突破，将符号函数的函数性质、几何特征与对应的复合算子的算子性质之间建立了密切的关系，这将会在拟共形映射和算子理论这两个分支之间建立沟通的桥梁。我们考虑了加权Bergman空间上的拟共形映射为符号的复合算子的性质，证明了郭坤宇教授他们的结果在更一般的条件下也成立。这方面的研究表明要进一步在算子理论的研究中发展调和分析的技巧。我们着重考虑了与之密切相关的regular权的加权调和Bergman空间上的Toeplitz算子的性质，同时在Toeplitz算子的totally Abelian性质、约化子空间等方面取得了积极的进展。