On the double commutant of Cowen–Douglas operators

Let T be a Cowen–Douglas operator. In this paper, we study the von Neumann algebra V ∗(T ) consisting of operators commuting with both T and T ∗ from a geometric viewpoint. We identify operators in V ∗(T ) with connection-preserving bundle maps on E(T ), the holomorphic Hermitian vector bundle associated to T . By studying such bundle maps, the structure of V ∗(T ) as well as information on reducing subspaces of T can be determined. © 2010 Elsevier Inc. All rights reserved.