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.