報告題目：On the flag structure for a class of Cowen-Douglas operators
報 告 人：紀奎 教授 河北師範大學
報告時間：2024年9月1日 8:20-9:00
報告地點：數學樓第一報告廳
校内聯系人：張遠航 zhangyuanhang@jlu.edu.cn
報告摘要：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.
報告人簡介：
紀奎，理學博士，河北師範大學數學科學學院，教授，博士生導師，中國數學會理事，河北省工業與應用數學學會副理事長，全國百篇優博獲得者，國家優青。主要從事算子理論的研究，主要關注複幾何在線性算子理論中的應用，研究内容包括Cowen-Douglas算子與Hermitian全純向量叢的結構與分類問題，包括利用幾何不變量刻畫算子的酉分類與相似分類、Cowen-Douglas 理論在C*-代數中的拓展與應用、算子的相似分類與Corona問題等。相關成果被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等數學期刊發表，并在應邀在2018年度、2023年度IWOTA（國際算子理論會議）作學術報告，主持國家自然科學基金4項、參與重點項目1項、國際重點合作項目1項、河北省傑出青年基金1項。曾獲2010年度全國百篇優秀博士論文獎勵、2013年度教育部自然科學二等獎（第二完成人）。
