Factorization of generalized holomorphic curve and homogeneity of operators

Abstract

The present paper concerns the homogeneity and similarity of operators in Cowen-Douglas class $B_n(\Omega )$. Let E be the Hermitian holomorphic vector bundle induced by $T\in B_n(\mathbb {D})$, and $E_{\alpha }$ be the Hermitian holomorphic vector bundle induced by $\phi _{\alpha }(T)$, where $\phi _{\alpha }$ is a M$\ddot{o}$bius transformation of the unit disk $\mathbb {D}$. Assume that the holomorphic Hermitian vector bundle $E_{\alpha }$ is congruent to $E\otimes \mathcal {L}_{\alpha }$ for some line bundle $\mathcal {L}_{\alpha }$ over $\mathbb {D}$, for each $\alpha \in \mathbb {D}$. Then it is shown that $\mathcal {L}_{\alpha }$ must be the trivial bundle and T is homogeneous. Furthermore, we investigate the similarity of operators with Fredholm index n associate with Hermitian holomorphic bundles. This characterization is given in terms of the factorization of generalized holomorphic curve induced by the corresponding holomorphic bundles.