Boundary Values and Cowen–Douglas Curvature

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