A local theory for operator tuples in the Cowen–Douglas class

Abstract We present a local theory for a commuting m -tuple S = ( S 1 , S 2 , ⋯ , S m ) of Hilbert space operators lying in the Cowen–Douglas class. By representing S on a Hilbert module M consisting of vector-valued holomorphic functions over C m , we identify and study the localization of S on an analytic hyper-surface in C m . We completely determine unitary equivalence of the localization and relate it to geometric invariants of the Hermitian holomorphic vector bundle associated to S . It turns out that the localization coincides with an important class of quotient Hilbert modules, and our result concludes its classification problem in full generality.