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.

本文涉及Cowen-Douglas类\（B_n（\ Omega）\）中算子的同质性和相似性。令E为\（T \ in B_n（\ mathbb {D}）\）中的Hermitian全纯矢量束，\（E _ {\ alpha} \）为\（\ phi _ { \ alpha}（T）\），其中\（\ phi _ {\ alpha} \）是单位磁盘\（\ mathbb {D} \）的M \（\ ddot {o} \） bius变换。假设该全纯埃尔米特向量丛\（E _ {\阿尔法} \）是全等\（E \ otimes \ mathcal {L} _ {\阿尔法} \）对于一些线束\（\ mathcal {L} _ {\ α }\）在\（\ mathbb {D} \）上，每个\（\ alpha \ in \ mathbb {D} \）中。然后表明\（\ mathcal {L} _ {\ alpha} \）必须是平凡的束，并且T是齐次的。此外，我们研究了与Hermitian全纯束相关的Fredholm指数n的算子的相似性。根据由相​​应的全纯束引起的广义全纯曲线的因式分解来给出该特征。