The numerical range of a contraction with finite defect numbers

An n-dilation of a contraction T acting on a Hilbert space H is a unitary dilation acting on H ⊕ C n . We show that if both defect numbers of T are equal to n, then the closure of the numerical range of T is the intersection of the closures of the numerical ranges of its n-dilations. We also obtain detailed information about the geometrical properties of the numerical range of T in case n = 1 .