On the classification and geometry for Cowen–Douglas operators of high-rank case

Abstract

In this paper we introduce a subclass of Cowen–Douglas operators of high-rank case denoted by ${\mathcal{FB}}_{n,1}(\Omega )$. Each operator $T\in {\mathcal{FB}}_{n,1}(\Omega )$ is induced by one Cowen–Douglas operator with rank $n$, another Cowen–Douglas operator with rank $1$, and an intertwining operator between them. By using this definition, we can construct plenty of Cowen–Douglas operators with high rank. By discussing the curvature of line bundle and second fundamental form of some rank $2$ bundle and its subbundle, we give the unitary classification of operators in ${\mathcal{FB}}_{n,1}(\Omega )$ and we reduce the number of unitary invariants of this kind of operators from $(n+1{)}^2$ to two.