Factorization of generalized holomorphic curve and homogeneity of operators

Abstract

The present paper concerns the homogeneity and similarity of operators in Cowen-Douglas class Bn(Ω). Let E be the Hermitian holomorphic vector bundle induced by T∈ Bn(D) , and Eα be the Hermitian holomorphic vector bundle induced by ϕα(T) , where ϕα is a Mo¨ bius transformation of the unit disk D. Assume that the holomorphic Hermitian vector bundle Eα is congruent to E⊗ Lα for some line bundle Lα over D, for each α∈ D. Then it is shown that Lα 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.