Operators of the Cowen–Douglas class with strong flag structure

Let \(\mathcal{F}\mathcal{B}_{n}(\Omega )\) $\mathcal{F}{\mathcal{B}}_n(\mathrm{\Omega })$ denote operators in the Cowen–Douglas class \(\mathcal {B}_{n}(\Omega )\) ${\mathcal{B}}_n(\mathrm{\Omega })$ possessing a flag structure. All the irreducible homogeneous operators in \(\mathcal {B}_{n}(\Omega )\) ${\mathcal{B}}_n(\mathrm{\Omega })$ belong to this class. The unitary invariants of this class of operators include the curvature and the second fundamental form of the corresponding line bundle. In this paper, we introduce a subclass of \(\mathcal{F}\mathcal{B}_{n}(\Omega )\) $\mathcal{F}{\mathcal{B}}_n(\mathrm{\Omega })$ 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 \(\mathcal {B}_{n}(\Omega )\) ${\mathcal{B}}_n(\mathrm{\Omega })$ 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 \(\frac{n(n-1)}{2}+1\) $\frac{n(n-1)}{2}+1$ to n. Moreover, we obtain a complete characterization of weakly homogeneous operators with large index and flag structure.