The classical Cowen-Douglas class of (commuting tuples of) operators possessing an open set of (joint) eigenvalues of finite constant multiplicity was introduced by Cowen and Douglas generalizing the backward shifts. Their unitary equivalence classes are determined by equivalence classes of certain hermitian holomorphic vector bundles associated to them on this set, and as emphasized in the work of Curto and Salinas, they are modelled by the adjoints of the multiplication operators by the independent variable(s) on a reproducing kernel Hilbert space.
Our goal is to develop a free noncommutative analogue of the Cowen-Douglas theory aiming to understand the notion of noncommutative vector bundles. We define the noncommutative Cowen–Douglas class using matrix joint eigenvalues as envisioned by Taylor and use the Taylor-Taylor series of free noncommutative function theory to show that the joint eigenspaces together constitute -- in a natural sense -- a noncommutative hermitian holomorphic vector bundle. If time permits, we also discuss noncommutative reproducing kernel Hilbert space models and noncommutative Gleason problem.
This is an ongoing work with Professor Victor Vinnikov.