In this repart, we analyze the intrinsic flag structure for operators in the Cowen--Douglas class, formulating the condition in terms of the associated holomorphic Hermitian vector bundle. We prove that the \mathcal{F}B_n operator class represents the exact subclass of Carlson--Clark extensions where the abstract module derivation reduces to a geometric chain rule. The rank-2 theory, based on a single extension cocycle, is developed in detail and then extended to the rank-r setting, where we prove that the level-wise strong intertwining condition automatically forces a ``crossed intertwining'' between consecutive cocycles---the operator-theoretic expression of the commutativity of mixed partial derivatives.

冀姗姗，河北经贸大学教师。研究方向集中于算子理论与复几何的交叉领域，主要关注Cowen-Douglas 算子组及相应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组的酉分类与相似分类、刻画算子组的弱齐次性。代表性科研论文发表于《Journal of Operator Theory》等学术期刊。