报告题目：On the flag structure for a class of Cowen-Douglas operators
报 告 人：纪奎 教授 河北师范大学
报告时间：2024年9月1日 8:20-9:00
报告地点：数学楼第一报告厅
校内联系人：张远航 zhangyuanhang@jlu.edu.cn
报告摘要：The explicit description of irreducible homogeneous operators in the Cowen–Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid, that is, the unitary equivalence class of the operator and the flag structure determine each other. A complete set of unitary invariants, which are somewhat more tractable than those of an arbitrary operator in the Cowen–Douglas class can be obtained. In this talk, we introduce a subclass of Cowen-Douglas operators which possesses a“strong" flag structure, and for which the curvature and the second fundamental form of the associated line bundle is a complete set of unitary invariants. We prove that this new class of operators is norm dense in the Cowen-Douglas class up to similarity. We obtain a classification modulo conjugation by an invertible operator for a large class of operators possessing a strong flag structure. Along the way, it is shown that the number of the similarity invariants found recently can be reduced from $\frac{n(n-1)}{2}+1$ to $n$. Moreover, we obtain a complete characterization of weakly homogeneous operators with large index and flag structure.
报告人简介：
纪奎，理学博士，河北师范大学数学科学学院，教授，博士生导师，中国数学会理事，河北省工业与应用数学学会副理事长，全国百篇优博获得者，国家优青。主要从事算子理论的研究，主要关注复几何在线性算子理论中的应用，研究内容包括Cowen-Douglas算子与Hermitian全纯向量丛的结构与分类问题，包括利用几何不变量刻画算子的酉分类与相似分类、Cowen-Douglas 理论在C*-代数中的拓展与应用、算子的相似分类与Corona问题等。相关成果被Advances in Mathematics、Journal of Functional Analysis、Israel Journal of Mathematics、Journal of Noncommutative Geometry、Journal of Operator Theory、Canadian Journal of Mathematics、Illinois Journal of Mathematics、European Journal of Mathematics、Studia Mathematica等数学期刊发表，并在应邀在2018年度、2023年度IWOTA（国际算子理论会议）作学术报告，主持国家自然科学基金4项、参与重点项目1项、国际重点合作项目1项、河北省杰出青年基金1项。曾获2010年度全国百篇优秀博士论文奖励、2013年度教育部自然科学二等奖（第二完成人）。
