If a bounded linear opertor T on a Hilbert space H lies in the Cowen-Douglas class, then its eigensheaf is locally free, but not conversely. We obtain a model for operators whose eigensheaves are locally free. We describe the eigensheaves for certain coanalytic Toeplitz operators, we show that the map from an operator to its eigensheaf is a functor from the category of bounded linear operators on Hilbert space to the category of Hilbert space-valued analytic sheaves, and we discuss relation between the eigensheaf of an operator and the sheaf that Putinar associates to an operator.

The work of the second author was supported through UGC SAP (DSA 1).