Operators of the Cowen–Douglas class with strong flag structure

Let F B n ( Ω ) denote operators in the Cowen–Douglas class B n ( Ω ) possessing a flag structure. All the irreducible homogeneous operators in B n ( Ω ) 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 F B n ( Ω ) 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 B n ( Ω ) 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 n ( n - 1 ) 2 + 1 to n . Moreover, we obtain a complete characterization of weakly homogeneous operators with large index and flag structure.