Boundary values and Cowen-Douglas curvature

We define a metric in terms of the Cowen-Douglas curvature for an operator T in B₁(Ω). Any boundary point of Ω that is a finite distance, with respect to this metric, from the eigenvalues in the interior is itself an eigenvalue of T. If T is represented as the adjoint of multiplication by the coordinate function on some holomorphic Hilbert space on Ω, this gives a condition under which functions in the space have limits along a path going to the boundary of Ω.