Towards a noncommutative theory of Cowen-Douglas class of noncommuting operators
Prahllad Deb, Victor Vinnikov
TL;DR
This work extends the classical Cowen–Douglas theory to tuples of noncommuting operators by developing a noncommutative (nc) Cowen–Douglas class and associating to each nc tuple a nc Hermitian holomorphic vector bundle. It leverages free nc function theory, cp nc reproducing kernel Hilbert spaces, and Taylor–Taylor analysis to build canonical models via adjoints of left multiplication operators on RKHS, while proving that unitary invariants are captured by the associated nc vector bundles. A major contribution is the nc Gleason problem, shown to be globally solvable with a unique solution on uniformly open, right-admissible nc domains, enabling a complete description of nc CD tuples as left-translation operators on cp nc RKHS. The results establish a foundation for noncommutative hermitian geometry, including vector bundles, cocycle data, and module-theoretic manifestations, with potential to inform further invariants and geometric structure in nc operator theory.
Abstract
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 the equivalence classes of certain hermitian holomorphic vector bundles associated with them on this set. This article develops a free noncommutative analogue of Cowen-Douglas theory to explore the notion of vector bundles in the setting of free noncommutative function theory. We define the noncommutative Cowen-Douglas class using matricial joint eigenvalues, as envisioned by Taylor, and show via the Taylor-Taylor series that the associated joint eigenspaces naturally form such a vector bundle, what we call a noncommutative hermitian holomorphic vector bundle. A key result is that the unitary equivalence class of a tuple in this class is completely determined by the equivalence class of its associated noncommutative vector bundle. This work lays the groundwork of the noncommutative hermitian geometry, which investigates noncommutative analogues of complex manifolds, vector bundles, and hermitian metrics by drawing on ideas from both complex hermitian geometry and operator theory. We also examine noncommutative reproducing kernel Hilbert space models and introduce the noncommutative Gleason problem, showing that elements of the noncommutative Cowen-Douglas class are essentially (up to unitary equivalence) adjoints of left multiplication operators by noncommuting independent variables in a noncommutative reproducing kernel Hilbert space.