The Cowen-Douglas operators with strongly flag structure

Denote $\mathcal{FB}_{n}(\Omega)$ as the collection of operators possessing a flag structure in the Cowen-Douglas class $\mathcal{B}_{n}(\Omega)$, and all the irreducible homogeneous operators in $\mathcal{B}_{n}(\Omega)$ belong to this class. G. Misra et al. pointed out in \cite{JJKM} that the unitary invariants of this class of operators include the curvature and the second fundamental form of the corresponding line bundle. In terms of the invariants, it is more tractable compared to general operators in $\mathcal{B}_{n}(\Omega)$. A subclass of $\mathcal{FB}_{n}(\Omega)$, denoted by $\mathcal{CFB}_{n}(\Omega)$, was proven to be norm dense in $\mathcal{B}_{n}(\Omega)$ in \cite{JJ}. In this paper, we introduce a smaller subclass of $\mathcal{FB}_{n}(\Omega)$ which possesses a strongly flag structure, and for which the curvature and the second fundamental form of the associated line bundle is a complete set of unitary invariants. And we notice that this class of operators is norm dense in $\mathcal{B}_{n}(\Omega)$ up to similarity. On this basis, we have completed the similar classification of a large class of operators with flag structure, which reduces the number of the similarity invariants in \cite{JKSX} from $\frac{n(n-1)}{2}+1$ to $n$. Furthermore, we also get a complete characterization of weakly homogeneous operators with high index and flag structure.